J>S>t   21  ifc 


4 


THE  LIBRARY  OF  THE 

UNIVERSITY  OF 

NORTH  CAROLINA 


PRESENTED  BY 

Robert  M.  Lester 


ROBERT    M.     LESTER 

PITTSBORO    ROAD 
CHAPEL   H,Ll-    N°R™   CAROL.NA 


•»    » 


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Digitized  by  the  Internet  Archive 

in  2012  with  funding  from 

University  of  North  Carolina  at  Chapel  Hil 


http://www.archive.org/details/elementaryalgebrOOdavi 


ELEMENTARY 


ALGEBRA: 


EMBRACING 


THE     FIRST    PRINCIPLES 


THE   SCIENCE. 


BY   CHARLES    DAVIE  S,   LL.D., 

AUTHOR     OF 

ARITHMETIC,   '  ELEMENTARY     GEOMETRY,    ELEMENTS     OF     SURVEYING, 

ELEMENTS   OF   DESCRIPTIVE   AND   ANALYTICAL   GEOMETRY,    EI.E- 

MENTS    OF   DIFFERENTIAL   AND   INTEGRAL   CALCULUS, 

AND   A   TREATISE   ON   SHADES,    SHADOWS, 

AND   PERSPECTIVE. 


NEW    YORK: 

PUBLISHED    BY   A.    S.    BARNES    &   CO. 

CINCINNATI  :    H.  W    DEKliY  &  CO. 

1857. 


A.    8.    BARNES    4    COMPANY  S    PUBLICATIONS. 
D  av  i  e  a'  Course  of  Mathematics, 

MATHEMATICAL    WORKS, 

IN  A   SERIES   OF  THKEE   PAET8  \ 

ARITHMETICAL,  ACADEMICAL,  AND  COLLEGIATE. 
BY  CHARLES  DAVIES,    LI.D. 

DAVIES'   LOGIC  AND  UTILITY    OF  MATHEMATICS. 

Tills  series,  combining  all  that  is  most  valuable  in  the  various  methods  of  Euro- 
pean instruction,  improved  and  matured  by  the  suggestions  of  more  than  thirty 
years'  experience,  now  forms  the  only  complete  consecutive  course  of  Mathematics. 
Its  methods,  harmonizing  as  the  works  of  one  mind,  carry  the  student  onward  by 
the  same  analogies  and  the  same  laws  of  association,  and  are  calculated  to  impart 
a  comprehensive  knowledge  of  the  science,  combining  clearness  in  the  several 
branches,  and  unity  and  proportion  in  the  whole  ;  being  the  system  so  iong  in  use 
at  West  Point,  through  which  so  many  men,  eminent  for  their  scientific  attainments, 
have  passed,  and  having  been  adopted,  as  Text  Books,  by  most  of  the  colleges  in  the 
United  States. 

I.    THE    ARITHMETICAL    COURSE   FOR    SCHOOLS 

1.  PRIMARY    ARITHMETIC    AND    TABLE-BOOK. 

2.  INTELLECTUAL    ARITHMETIC. 

3.  school  arithmetic.     (Key  separate.) 

4.  GRAMMAR    OF    ARITHMETIC. 

II.     THE    ACADEMIC    COURSE. 

1.  the  university  arithmetic.     (Key  separate.) 

2.  PRACTICAL  mathematics   FOR   practical  men. 

3.  elementary  algebra.     (Key  separate.) 

4.  elementary  geometry  and  trigonometry. 

5.  elements  of  surveying. 

iii.    the   collegiate  course. 

1.  davies'  bourdon's  algebra. 

2.  davies'  legendre's  geometry  and  trigonometry. 

3.  davies'  analytical  geometry. 

4.  da  vies'  descriptive  geometry. 

5.  davies'  shades,  shadows,  and  perspective. 

6.  davies'  differential  and  integral  calculus. 


Entered  according  to  the  Act  of  Congress,  in  the  year  one  thousand  eight  hundred 
and  fifty-two,  by  Charles  Davies,  in  the  Clerk's  Office  of  the  District  Court  of  the 
United  States,  for  the  Southern  District  of  New  York. 


A    P.   JONES    AMP   CO.     8TXBEOTYT1I10. 


PREFACE, 


Although  Algebra  naturally  follows  Arithmetic  in 
a  course  of  scientific  studies,  jet  the  change  from  the 
methods  of  reasoning  on  numbers  to  a  system  of  rea- 
soning entirely  conducted  by  letters  and  signs,  is  rather 
abrupt  and  not  unfrequently  discourages  the  pupil. 

In  this  work,  it  has  been  the  intention,  to  form  a 
connecting  link  between  Arithmetic  and  Algebra,  to 
unite  and  blend,  as  far  as  possible,  the  reasoning  on 
numbers  with  the  more  abstruse  methods  of  analysis. 

The  Algebra  of  M.  Bourdon  has  been  closely  follow- 
ed. Indeed,  it  has  been  a  part  of  the  plan,  to  furnish 
an  introduction  to  that  admirable  treatise,  which  is 
justly  considered,  both  in  Europe  and  this  country,  as 
the  best  work  on  the  subject  of  which  it  treats,  that 
has  yet  appeared.  The  work  of  Bourdon,  however, 
even  in  its  abridged  form,  is  too  voluminous  for  schools, 
and  the  reasoning  is  too  elaborate  and  metaphysical 
for  beginners. 

It  has  been  thought  that  a  work  which  should  so  far 
modify  the  system  of  M.  Bourdon  as  to  bryig  it  within 
the  scope  of  our  common  schools,  by  giving  to  it  a 
more  practical  and  tangible  form,  could  not  fail  to  be 
useful.  Such  is  the  object  of  the  Elementary 
Algebra. 


iv  PHEFACS. 

The  success  which  has  attended  this  effort,  so  to 
simplify  the  subject  of  Algebra  as  to  bring  it  within 
the  range  of  common  school  instruction,  has  been  pecu- 
liarly gratifying.  It  is  about  twelve  years  since 
the  first  publication  of  the  Elementary  Algebra. 
Within  that  time,  between  twenty  and  thirty  editions 
have  been  printed,  and  several  works  of  other  authors, 
have  also  appeared,  modelled  after  the  same  general  plan, 

In  the  present  edition,  few  alterations  have  been 
made  in  the  general  plan  of  the  work.  The  introduc- 
tion has  been  somewhat  enlarged  for  the  purpose  of 
preparing  the  pupil  by  a  thorough  system  of  mental 
training,  for  those  processes  of  reasoning  which  are 
peculiar  to  the  algebraic  analysis. 

I  have  availed  myself,  in  the  present  edition,  of 
many  valuable  suggestions  from  teachers  who  have 
used  the  work,  and  favored  me  with  their  opinions 
both  of  its  defects  and  merits. 

The  criticisms  of  those  engaged  in  the  daily  business 
of  teaching  are  invaluable  to  an  author;  and  I  shall 
feel  myself  under  special  obligations  to  all  such  who 
will  be  at  the  trouble  to  communicate  to  me,  at  any 
time,  such  changes,  either  in  methods  or  language,  as 
their  experience  may  point  out.  It  is  only  through 
the  cordial  co-operation  of  teachers  and  authors — by 
joint  labors  and  mutual  efforts,  that  the  text-books  of 
the  country  can  be  brought  to  any  reasonable  degree 
of  perfection. 

Fishkill  Landing,  ) 
July,  1852.      j 


I 


CONTENTS. 


CHAPTER    I. 


PRELIMINARY    DEFINITIONS   AND   REMARKS. 

■A.RT1CLE0. 

Algebra — Definitions — Explanation  of  the  Algebraic  Signs,     -  1 — 23 

Similar  Terms — Reduction  of  Similar  Terms,          ...  23 — 26 

Addition — Rule,    -                                    26— -28 

Subtraction — Rule — Remark,          ......  28 — 33 

Multiplication — Rule  for  Monomials,       .....  83 — 36 

Rule  for  Polynomials  and  Signs,     ......  86 — 38 

Remarks — Properties  proved,         .......  88 — 42 

Division  of  Monomials — Rule,         .......  42 — 45 

Signification  of  the  Symbol  a0,                         ....  45—  45 

Of  the  Signs  in  Division,        --.-...  46—47 

Division  of  Polynomials,         .......  47 — 4.9 


CHAPTER   II. 


ALGEBRAIC     FRACTIONS. 


Definitions — Entire  Quantity — Mixed  Quantity, 

To  Reduce  a  Fraction  to  its  Simplest  Terms,  - 

To  Reduce  a  Mixed  Quantity  to  a  Fraction,     - 

To  Reduce  a  Fraction  to  an  Entire  or  Mixed  Quantity, 

To  Reduce  Fractions  to  a  Common  Denominator,     - 

To  Add  Fractions, 


49—52 
52 
53 
54 
55 
56 


VI 


COHTKKT8 


To  Subtract  Fractions, 
To  Multiply  Fractions, 
To  Divide  Fractions, 


ARTICLIS. 

57 
58 
59 


CHAPTER    III 


EQUATIONS    OF   THE    FIEST    DEGEE3. 


Definition  of  an  Equation — Properties  of  Equations,       -        -  60 — 68 

Transformation  of  Equations — First  and  Second,     -         -         -  66 — 70 

Resolution  of  Equations  of  the  First  Degree — Rule,        -         -  70 

Questions  involving  Equations  of  the  First  Degree,  -  -  71 — 72 
Equations   of    the    First    Degree    involving   Two   Unknown 

Quantities,      ---..- 72 

Elimination — By  Addition — By  Subtraction — By  Comparison,  73 — 76 
Resolution  of  Questions    involving  Two  or  more   Unknown 

Quantities,     -                 76—79 

CHAPTER    IV. 

OF    POWEES. 


Definition  of  Powers,     - 
To  raise  Monomials  to  any  Power, 
To  raise  Polynomials  to  any  Power, 
To  raise  a  Fraction  to  any  Power,  - 
Binomial  Theorem, 


79 

80 

81 

32—83 

84—90 


CHAPTER   V. 


Definition    of    Squares — Of    Square    Roots — And    Perfect 

Squares, '  -  90—96 

Rule  for  Extracting  the  Square  Root  of  Numbers,       -        -  96 — 100 

Square  Roots  of  Fractions, 100 — 103 

Square  E,oots  of  Monomials, 103 — 107 

Calculus  of  Radicals  of  the  Second  Degree,         -         -         -  107 — 109 


CONTENT*.. 


Vll 


ARTICLES. 

Addition  of  Radicals, •  109 

Subtraction  of  Radicals,      ..*--•«  no 

Multiplication  of  Radicab           ......  m 

Division  of  Radicals,  -                  ......  112 

Extraction  of  the  Square  Root  of  Polynomials,  -                  -  113 — 116 

CHAPTER    VI 


Equations  of  the  Second  Degree, 

Definition  and  Form  of  Equations, 

Incomplete  Equations, 

Complete  Equations, 

Four  Forms,       .... 

Resolution  of  Equations  of  the  Second  Degree, 

Properties  of  the  Roots,     .... 


116 

116—118 
118—122 
122 
123—121 
127—128 
128—134 


CHAPTER   VII. 


Of  Progressions,         ..... 

Progressions  by  Differences,        ... 

Last  Term, 

Sum  of  the  Extremes — Sum  of  the  Series, 

The  Five  Numbers — To  Find  any  Number  of  Means, 

Geometrical  Proportion  and  Progression, 

Various  Kinds  of  Proportion,      ... 

Geometrical  Progression,     .... 

Last  Term — Sum  of  the  Series, 

Progressions  having  an  Infinite  Number  of  Terms, 

The  Five  Numbers — To  Find  One  Mean,     - 


135 
136—138 

138—140 
140—141 
141—144 

144 
144—166 

166 
167—171 
171—172 
172—173 


CHAPTER    VIII. 


Theory  of  Logarithms, 


174—179 


SUGGESTIONS   TO   TEACHERS. 


1.  The  Introduction  is  designed  as  a  mental  exercise.  If 
thoroughly  taught,  it  will  train  and  prepare  the  mind  of  the 
pupil  for  those  higher  processes  of  reasoning,  which  it  is  the 
peculiar  province  of  the  algebraic  analysis  to  develop. 

2.  The  statement  of  each  question  should  be  made,  and 
every  step  in  the  solution  gone  through  with,  without  the 
aid  of  a  slate  or  black-board ;  though,  perhaps  in  the  be- 
ginning, some  aid  may  be  necessary  to  those  unaccustomed 
to  such  exercises. 

3.  Great  care  must  be  taken  to  have  every  principle  on 
which  the  statement  depends,  carefully  analyzed  ;  and  equal 
care  is  necessary  to  have  every  step  in  the  solution  distinct- 
ly explained. 

4.  The  reasoning  process  embraces  the  proper  connection 
of  distinct  apprehensions,  and  the  consequences  which  follow 
from  such  a  connection.  Hence,  the  basis  of  all  reasoning 
must  lie  in  distinct  elementary  ideas. 

5.  Therefore,  to  teach  one  thing  at  a  time — to  teach  that 
thing  well — to  explain  its  connections  with  other  things,  and 
the  consequences  which  follow  from  such  connections,  would 
seem  to  embrace  the  whole  art  of  instruction. 


INTRODUCTION. 


LESSON   I, 


1.  John  and  Charles  have  twelve  apples  between  tneou 
and  each  has  as  many  as  the  other :  how  many  has  each  ? 

If  we  suppose  the  apples  divided  into  two  equal  parts,  it 
is  plain  that  John  will  have  one  part  and  Charles  the  other : 
hence,  they  will  each  have  six  apples. 

In  Algebra,  we  often  represent  numbers  by  the  letters  of 
the  alphabet ;  that  is,  we  take  a  letter  to  stand  for  a  num- 
ber. Thus,  let  x  stand  for  the  apples  which  John  has. 
Then,  as  Charles  has  an  equal  number,  x  will  also  stand  for 
the  apples  which  he  has.  But  together,  they  have  twelve 
apples,  hence,  twice  x  must  be  equal  to  12.  This,  we  write 
thus 

*  +  «  =  2»-=;i2; 

and  if  twice  x  is  equal  to  12,  it  follows  that  once  x,  or  x, 
will  be  equal  to  12  divided  by  2,  or  equal  to  6.  This,  we 
write  thus : 

When  we  write  x  by  itself,  we  mean  one  #,  or  the  same  as 
la;.  If  we  write  2ar,  we  mean  that  x  is  taken  twice ;  if  3ar, 
that  it  is  taken  three  times,  &c. 

1.  In  the  first  question,  how  many  apples  has  each  boy?  By  what 
arc  numbers  represented  in  Algebra  ?  If  x  stands  by  itself,  how  many 
times  x  are  expressed  ?  What  does  2x  denote  ?  What  3a;  ?  What  4x. 
«fec.  If  we  have  x  +  x,  to  how  many  times  x  is  it  equal  ?  If  we  hav* 
the  value  of  2a;,  how  do  we  find  the  value  of  x  f 

1 


'£  ELEMENTARY     ALGEBRA. 

2.  James  and  John  together  have  24  peaches,  and  one  has 
as  many  as  the  other  :  how  many  has  each  1 

Let  x  stand  for  the  number  of  peaches  which  James  has : 
then  x  will  also  denote  the  number  of  peaches  which  John 
has ;  and  since  they  have  24  between  them, 
x  +  x  —  24  ; 

24 

that  is,         2x  =  24     and     x  =  —  =  12. 

Therefore,  each  has  twelve  peaches. 

1  William  and  John  have  86  pears,  and  one  has  as  many 
as  tne  other  :  how  many  has  each  1 

Let  the  number  which  each  has  be  denoted  by  x. 
men  x  +  x  =  36 ; 

that  is,         2x  =  36     and     x  =  —  =  18. 

m 

4.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  20 1 

Let  the  number  be  denoted  by  x  :  then,  as  the  number  is 
to  be  added  to  itself,  we  have 

x  +  x  =  20  ; 

20 

that  is,  2x  =  20     or     x  =  —  =  10. 

m 

Hence,  10  is  the  number. 

5.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  30  ? 

2 1  In  the  second  question,  what  does  x  stand  for?  What  is  twice  x 
equal  to  ?     How  then  do  you  find  the  value  of  x  ? 

3.  In  the  third  question,  what  does  x  stand  for  ?  What  is- x  equal  to! 
How  do  you  find  the  value  of  x  ? 

4i  In  the  fourth  question,  what  does  x  stand  for  ?  What  is  twice  x 
equal  to  ?     How  do  you  then  find  x  ? 

5.  In  the  fifth  question,  what  does  x  stand  for  ?  How  do  you  find  iti 
value  ? 


INTRODUCTION  .  3 

6.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  50 ? 

7.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  100? 

8.  What  number  added  to  itself  will  give  a  sum  equal 
to  80? 

9.  What  number  added  to  itself  will  give  a  sum  equal 
to  25? 

10.  What  number  added  to  itself  will  give  a  sum  equal 
to3P4? 


LESSON   II. 

)  John  and  Charles  together  have  12  apples,  and  Charles 
h»  i  twice  as  many  as  John  :  how  many  has  each  ? 

If  we  now  suppose  the  apples  to  be  divided  into  three 
equal  parts,  it  is  evident  that  John  will  have  one  of  the 
parts  and  Charles  two  of  them. 

Let  us  denote  by  z  the  number  of  apples  which  John  has. 

Then  2x  will  denote  what  Charles  has,  and  x  +  2.r  will  be 

equal  to  the  whole  number  of  apples.    This  equality  is  thus 

expressed  : 

z+  2x  =  12; 

12 

that  is,  3x  =  12     or     x  =  —  =  4, 

o 

therefore,  John  has  4  apples,  and  Charles  8. 

6 1  How  do  you  find  the  value  of  x  in  the  6  th  question  ?  How  in  the 
8th  ?     How  in  the  9th  ?     How  in  the  10th  ? 

Questions  on  Lesson  II. — 1<  Into  how  many  parts  do  we  suppose 
the  1 2  apples  to  be  divided  ?  How  many  of  the  parts  will  John  have  f 
What  is  the  value  of  each  part  ?  If  a;  stands  for  one  of  the  parts,  what 
will  stand  for  two  parts  ?  What  for  three  parts  ?  If  you  have  the 
value  of  Zse,  bow  will  you  Cud  the  value  of  x  ? 


4  ELEMENTARY     ALGEBRA. 

2.  James  and  John  have  30  pears,  and  John  has  twice  as 
many  as  James  :  how  many  has  each  1 

Here,  again,  let  us  suppose  the  whole  number  to  be  divided 
into  three  equal  parts,  of  which  James  must  have  one  part, 
and  John  two. 

Let  us  then  denote  by  x,  the  number  of  pears  which 
James  has :  then  2x  will  denote  the  number  of  pears  which 
John  has,  and  x  -j-  2x  will  be  equal  to  the  whole  number  of 
pears  :  and  we  shall  have 

x  +  2x  =  30  ; 

,       •  «,         «*  30 

that  is,  Sx  =  30     or     x  =  —  =  10. 

o 

3.  William  and  John  have  48  quills  between  them,  and 
John  has  twice  as  many  as  William  :  how  many  has  each? 

Let  the  number  of  quills  which  William  has  be  denoted 
by  x  :  then,  since  John  has  twice  as  many,  his  will  be  de- 
noted by  2x,  and  the  number  possessed  by  both,  will  be  de- 
noted  by  x  -j-  2x.     Hence,  we  shall  have 
x  -\-2x  =  48 ; 

48 
that  is,  ox  =  48     or     x  z=  —  =  16. 

& 

Hence,  William  has  16  quills,  and  John  32. 

4.  What  number  is  that  which  added  to  twice  itself,  will 
give  a  sum  equal  to  60? 

Let  the  number  sought  be  denoted  by  x,  then  twice  the 
number  will  be  denoted  by  2x,  and  we  shall  have 

z  -f 2x  =  60 ; 

60 
that  is,  3x  =  60     or     x  =  —  =  20  ; 

o 

and  we  see  that  20  added  to  twice  itself  will  give  60. 

1,  In  question  second,  what  is  the  value  of  one  of  the  parts  ?  What 
in  question  3d  ?     How  do  you  state  question  4th  ? 


INTRODUCTION.  5 

5.  John  says  to  Charles,  "give  me  your  marbles  and  I 
shall  have  three  times  as  many  as  I  have  now."  "  No," 
says  Charles,  "  but  give  me  yours,  and  I  shall  have  just  51." 
How  many  had  each  % 

Let  the  number  of  marbles  which  John  has  be  denoted 

by  x :  then,  2x  will  denote  the  number  which  Charles  has, 

and  since  they  have  51  in  all,  we  write 

x  +  2x  =  51 ; 

51 
that  is,  3x  =  51     or     x  =  — -  =  17. 

o 

6.  What  number  is  that  which  added  to  twice  itself  will 
give  a  sum  equal  to  75  1 

Let  the  number  be  denoted  by  x :  then,  twice  the  number 

will  be  expressed  by  2x,  and 

x  +  2x  =  75  ; 

75 
that  is,  3a;  =  75     or    x  =  —  =  25. 

o 

7.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  90  1 

8.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  57  l 

9.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  39  1 

10.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  21 1 


LESSON   III. 
1.  If  James  and  John  together  have  24  quills,  and  Joho 
has  three  times  as  many  as  James,  how  many  has  each?  , 

5.  How  do  you  state  question  5th  ?  Explain  the  6th  question  ?  Also 
the  7th  ?  What  is  the  required  number  in  the  8th  ?  What  in  the  9th  • 
What  in  the  10th  ? 


6  ELEMENTAltT     A  L  6  E  B  K  A  . 

It  is  plain  that  if  we  suppose  the  twenty-four  quills  to  be 
divided  in  four  equal  parts,  that  James  will  have  one  of  the 
parts,  and  John  three. 

Let  us  now  denote  by  x  the  number  of  quills  which  James 
has  :  then  3x  will  denote  the  number  of  quills  which  John 
has,  and  we  shall  have 

x  +  3«  =  24  ; 

24 

that  is,  4r  =  24     or     x  =  —  =  6. 

2.  What  number  is  that  which  added  to  three  times  itself 

will  give  a  sum  equal  to  48  % 

If  we  denote  the  number  by  x,  we  shall  have 

*  +  3x  =  48  ; 

48 
that  is,  4a;  =  48     or     x  =  — -  =  12. 

4 

8.  John  and  Charles  have  60  apples  between  them,  and 
Charles  has  thra<?  times  as  many  as  John :  how  many  has  each? 

If  we  suppose  the  number  of  apples  to  be  divided  into 
four  equal  parts,  it  is  evident  that  John  will  have  one  of 
those  parts,  and  Charles  three. 

Let  x  =  the  number  which  John  has ;  then  3a:  will  stand 

for  the  number  which  Charles  has,  and  we  shall  have 

x  -|-  ox  =  60  ; 

,  60 

that  is,  4x  =  60     or     x  =  —  =  15. 

4 

Hence,  John  will  have  15  and  Charles  45. 

li  If  the  twenty-four  quills  be  divided  into  four  equal  parts,  how 
many  parts  will  John  have  ?  How  many  will  James  have  ?  What  is 
each  part  equal  to  ? 

2.   If  three  times  a  number  be  added  to  the  number,  how  many  times   . 
will  the  number  be  taken  ?     If  Ax  is  equal  to  48,  what  is  the  value  of  x  ? 
Explain  the  third  question.      If  Ax  is  equal  to  60,  how  do  you  find  the 
value  of  js  ? 


INTRODUCTION.  7 

4.  What  number  is  that  which  being  added  to  three  times 
itself  will  give  a  sum  equal  to  100  % 

Let  the  number  be  denoted  by  x :  then 
x  +  ox  —  100  ; 

that  is,  4x  =  100    or  'x  = =  25. 

4 

5.  What  aumber  is  that  which  if  added  to  four  times  it- 
self, the  sum  will  be  equal  to  60  1 

Let  x  denote  the  number.     Then, 

x  -f  Ax  =  60  ; 

60 
that  is,  5x  =  60    or    x  =—-  =  12. 

5 

6.  What  number  is  that  which  being  multiplied  by  3,  and  the 
product  added  to  twice  the  number  will  give  a  sum  equal  to  75 1 

Let  the  number  be  denoted  by  x. 
Then,     ox  =  the  product  of  the  number  by  3 ; 
and        2x  =  twice  the  number  ; 
and  ox  -f-  2x  =  5x  =  75  ; 

75 

or  x  =  —  =  15,  the  required  number. 

o 

7.  What  number  is  that  which  being  added  to  three  times 
itself  will  give  a  sum  equal  to  140  1 

8.  What  number  is  that  which  being  multiplied  by  5,  and 
the  product  added  to  the  number,  will  give  a  sum  equal  to  2-10 1 

9.  What  number  is  that  which  being  multiplied  by  2,  and 
then  by  3,  and  the  products  added,  will  give  125  ? 

5.  If  a  number  be  added\t!a  four  times  itself,  how  many  times  will  the 
number  be  taken  ? 

6.  If  x  stands  for  any  numbgrj"  w-bSfcjvill  stand  for  three  times  that 
number  ?     What  for   twice  the  number  ?     Explain  the  7th  question 
How  do  you  state  it?     What  is  4x  equal  to  ?     Why?     How  then  do 
you  find  x  ?    How  do  you  state  the  8th  question  ?    What  is  6x  equal  to  ? 
How  then  do  you  find  x  ? 

9.  If  x  denotes  a  number,  what  will  stand  for  twice  the  number! 
What  fur  threu  times  the  number  I 


ELEMENTARY      ALUEDIU. 


LESSON    IV. 

1.  John  and  Charles  together  have  80  apples,  and  Chan  id 
has  four  times  as  many  as  John  :  how  many  has  each  1 

If  we  suppose  the  80  apples  to  be  divided  into  5  equal 
parts,  it  is  evident  that  John  will  have  one  of  the  parts,  and 
Charles  four. 

Let  x  stand  for  the  number  of  apples  which  John  haa  : 

then  4x  will  stand  for  the  number  which  Charles  has  ;  and 

x  +  4x  =  80  ; 

80 
that  is,  bx  =  80     and    x  =  —  ==  16. 

5 

2.  What  number  added  to  four  times  itself  will  give  a  sum 
equal  to  90  ? 

3.  What  number  added  to  five  times  itself  will  give  a  sum 
equal  to  120  ? 

4.  What  number  added  to  six  times  itself  will  give  a  sum 
equal  to  245  1 

5.  What  number  added  to  seven  times  itself  will  give  a 
sum  equal  to  860  ? 

6.  What  number  added  to  five  times  itself  will  give  a  sum 
equal  to  200  1 

7.  What  number  added  to  itself  and  the  sum  to  four 
times  the  number  will  give  a  sum  equal  to  72  ? 

1.  If  a;  stands  for  John's  apples,  what  will  denote  Charles'  ?  What, 
Will  stand  for  the  apples  which  they  both  have  ?  If  5x  is  equal  to  80, 
what  will  x  be  equal  to  ?  If  a  number  be  added  to  four  times  itself, 
how  many  times  will  the  number  be  taken  ?  If  5  times  a  number  ia 
equal  to  90,  what  is  the  value  of  the  number?  Explain  example  3d 
Explain  question  4th.  What  doe3  x  stand  for?  Explain  the  5th  quea 
lion.     Explain  example  0th. 


INTRODUCTION.  9 


LESSON   V. 

1.  What  number  added  to  five  times  itself,  will  give  a 
sum  equal  to  60  ? 

2.  John  has  a  number  of  marbles  and  buys  four  times  as 
many  more,  when  he  has  seventy -five :  how  many  had  he 
at  first  ? 

3.  If  x  be  taken  seven  times,  and  then  eight  times,  how 
many  times  will  it  be  taken  in  all  ? 

4.  If  x  be  made  equal  to  5,  in  the  last  example,  what  wilf 
be  the  numerical  value  of  the  sum  1 

5.  Find  two  numbers  whose  sums  shall  be  fifty,  and  one 
of  them  four  times  the  other  ] 

Let  x  denote  the  less  number  : 
then,  4x  will  denote  the  greater : 
and  by  the  conditions  of  the  question 
x  +  4:X  =  50  ; 

hence,  5x  =  50  ;     or    'x  =  —  =  10. 

5 

6.  Find  two  numbers  whqse  sum  shall  be  forty-five,  and 
one  of  them  eight  times  the  other. 

7.  Divide  the  number  thirty  into  two  such  parts  that  the 
greater  shall  be  four  times  the  less. 

8.  Divide  the  number  forty-eight  into  two  such  parts  that 
the  greater  shall  be  five  times  the  less. 

9.  Divide  the  number  sixty-four  into  two  such  parts  that 
the  greater  shall  be  seven  times  the  less. 

10.  What  is  the  sum  of  9x  and  three  xl  What  is  the 
gum  equal  to,  numerically,  if  x  is  equal  to  5  % 

11.  What  is  the  sum  of  eight  x  and  one  x  1  What  is  the 
sum  equal  to,  numerically,  when  x  is  7? 

1* 


10  ELEMENTARY     ALGEBRA. 

12.  What  is  the  sum  of  x  +  x  +  3x  +  Ax  4-  bx  ?  What 
is  this  sum  equal  to,  numerically,  if  x  is  2  ? 

13.  What  is  the  sum  of  2.r  +  a;  +  3.r  +  4a;  4-  x  +  a:  ] 
What  is  the  sum  equal  to,  numerically,  when  a;  is  9  1 

14.  James  and  John  wish  to  share  thirty-six  apples,  so 
that  James  shall  have  three  times  as  many  as  John  :  how 
many  will  each  have  ? 

15.  What  number  added  to  eight  times  itself,  and  this 
sum  to  three  times  the  number,  will  give  a  sum  equal  to  48  ? 

16.  What  number  is  that  whose  ninth  part  added  to  the 
number  will  give  a  sum  equal  to  twenty  ? 

Let  the  number  be  denoted  by  9x : 
then  one-ninth  of  9x  will  be  denoted  by  x ;  and  by  the  con 
ditions  of  the  question 

9x  +  x  =  20, 

20 
hence,  10a;  =  20     or     x  =  — -  =  2. 

Then,  if  x  =  2,    9a;  =  18,  the  number  sought. 


LESSON    VI. 

1.  What  number  added  to  six  times  itself,  and  then  to 
five  times  itself,  will  give  a  sum  equal  to  twentv-four  1 

Let  x  denote  the  number  : 
then,  6a;  =     six  times  the  number, 

and  bx  =     five  times  the  number : 

and  by  the  conditions  of  the  question 
x  -f-  6a;  +  5a;  =  24 : 

24 
hence,  12a;  =  24     or     x  —  —  =  2. 

\4& 


INTRODUCTION.  11 

2.  What  number  added  to  twice- itself,  then  to  three  times 
itself,  to  four  times  itself,  and  to  five  times  itself,  will  give 
a  sum  equal  to  fifteen  ? 

S.  Divide  twenty-one  into  three  such  parts,  that  the  second 
shall  be  equal  to  four  times  the  first,  and  the  third  to  four 
times  the  second. 

4.  A  farmer  has  three  times  as  many  sheep  as  goats,  and 
one-third  as  many  lambs  as  sheep  :  he  has  thirty  in  all : 
how  many  has  he  of  each  sort  1 

Let  x     denote  the  number  of  goats  : 

then  3a;     will  denote  the  number  of  sheep, 

and  x     will  denote  the  number  of  lambs  ; 

and  by  the  conditions  of  the  question 
x  +  2>x  +  x  =  30     the  number  in  all. 

Then  5x  =  30  ;  or  x  =  —  =  6,     the  lambs  or  goats. 

o 

Also,  3#  =  3x6  =  18    the  number  of  sheep. 

5.  James  has  twice  as  many  dime-pieces  as  cent-pieces, 
and  has  forty-two  cents  in  all :  how  many  dime-pieces  has 
he? 

6.  John  has  two  sisters  and  one  brother,  and  wishes  to 
divide  thirty  dollars  between  them.  He  wishes  to  give  the 
elder  sister  twice  as  much  as  the  younger,  and  the  brother 
as  much  as  both  the  sisters  :  how  much  must  he  give  to 
each  1 

7.  An  orchard  contains  thirty-five  trees.  There  is  an 
equal  number  of  plum  trees  and  pear  trees  ;  but  there  are 
three  times  as  many  cherry  trees  as  plum  trees,  and  twice 
oa  many  apple  trees  as  pears :  how  many  of  each  sort  ? 


12  ELEMENTARY     ALGEBRA. 

8.  Divide  twenty-four  mto  three  such  parts  that  the 
second  shall  be  double  the  first,  and  the  third  three  times 
the  first. 

9.  Divide  the  number  fourteen  into  three  such  parts,  that 
the  second  shall  be  double  the  first,  and  the  third  double 
the  second. 

10.  John  has  three  times  as  many  marbles  as  William 
has  tops :  the  tops  cost  three  cents  a  piece,  and  the  marbles 
one  cent,  and  together  they  cost  thirty  cents :  howruany 
had  each? 

-11.  Jane  has  a  blush  rose  bush,  a  moss  rose  bush,  and  a 
white  rose  bush  ;  and  together  they  have  thirty-three  buds ; 
the  buds  on  the  second  are  double  those  on  the  first,  and 
those  on  the  third  four  times  those  on  the  second :  how 
many  on  each"? 


LESSON    VII. 

1.  James  has  three  times  as  m&ny  marbles  as  Charles, 
and  together  they  have  thirty-two  :  how  many  has  each  ? 

Let     x  =  the  number  of  marbles  which  Charles  has : 
then      Sx  =   the  number  James  has  : 
and         x  +  Bx  =  32  what  both  have. 

32 

Then  4s  =  32  ;    or    x  =  —  =  8  : 

4 

therefore,  Charles  has  8,  and  James  8x3=  24. 

2.  John,  Charles,  and  "William  have  ninety  books  :  Charles 
has  five  times  as  many  as  John,  and  William  four  times  as 
many  as  John  :  how  many  has  each  1 


INTRODUCTION.  13 

Let     x  =   the  number  which  John  has : 
then      5x  =   the  number  Charles  has, 
and       4x  =   the  number  William  has. 
Then,     x  -f-  5x  +  4x  =  90,  the  number  they  all  have. 

90 

Hence,  10z  =  90  :    or    x  —  — -  =  9. 

10 

Therefore,  John  has  9 ;  Charles  45,  and  William  36. 

3.  The  sum  of  three  number  is  twenty-four :  the  second 
is  twice  the  first,  and  the  third  five  times  the  first :  what 
are  the  numbers  % 

4.  The  sum  of  three  numbers  is  thirty-eight :  the  second 
is  three  times  the  first,  and  the  third  five  times  the  second  : 
what  are  the  numbers  ? 

5.  The  sum  of  three  numbers  is  forty-eight :  the  second  i3 
seven  times  the  first,  and  the  third  is  equal  to  the  sum  of 
the  first  and  second  :  what  are  the  numbers  1 

6.  The  sum  of  four  numbers  is  seventy  :  the  second  is 
four  times  the  first ;  the  third  three  times  the  first,  and  the 
fourth  double  the  third :  what  are  the  numbers  1 

7.  Divide  the  number  thirty-nine  into  three  such  parts, 
that  the  second  shall  be  three  times  the  first,  and  the  third 
three  times  the  second. 

8.  Divide  the  number  seventy-five  into  two  such  parts, 
that  the  less  shall  be  one-fourth  of  the  greater. 

9.  Divide  one  hundred  and  thirty-three  into  three  such 
parts,  that  the  second  shall  be  three  times  the  first,  and  the 
third  five  times  the  second. 

10.  Divide  eighty -five  into  four  such  pasts  that  the  second 
shall  be  four  times  the  first,  the  third  four  times  the  second 
and  the  fourth  four  times  the  third. 


14  ELEMENT AKY     ALGEUUA. 


LESSON   VIII. 

Note. — Let  the  pupil  now  read  Articles  60,  61,  and  that  part  of  Art 
64,  which  is  found  on  page  90.     Also,  Art.  65. 

1.  James  receives  five  apples  from  John,  and  then  has 
twelve  :  how  many  had  he  at  first  % 

Let     x  =  the  number  he  had  at  first. 
Then       x  -f-  5  =  12,  what  he  had  afterwards. 

Now,  if  x  increased  by  5,  equals  12,  x  must  be  less 
than  12,  by  5.     Hence, 

x  =  12  —  5  =  7. 

When  we  take  a  number  from  one  member  of  an  equa- 
tion and  place  it  in  the  other,  we  are  said  to  transpose  it. 

2.  William  has  eight  marbles  more  than  John,  and  to- 
gether they  have  thirty-six :  how  many  has  each  ] 

Let  x  =         the  number  which  John  has : 

then     x  -f-  8  =         the  number  William  has, 
and    2x  -f-  8  =  36,  the  number  they  both  have. 

Now,  if  2x  increased  by  8  is  equal  to  86,  2*  must  be 
equal  to  86  diminished  by  8 :   hence, 

2x  =  36  —  8  =  28 

or  x  =  — -  =  14. 

2 

Hence,  we  see  that  a  plus  number  may  be  transposed  from 
one  member  of  an  equation  to  the  other,  by  simply  chang 
ing  its  sign  to  minus. 

3.  A  father's  age  is  double  his  son's,  and  the  sum  of 
their  ages  increased  by  four  is  equal  to  64 :  what  is  the  age 
of  each  1 


INTRODUCTION.  15 

Let        x  denote  the  son's  age  : 
then         2x  will  denote  the  father's  age, 
and  2x  +  x  will  denote  the  sum  of  their  ages. 

But  by  the  conditions  of  the  question 
2x  +  x  +  4  =  64  ; 
hence,  3x  +  4  =  64 

and  Sx  =  64  -  4  =  60, 

PCi 

or  £  =  —  =  20,  the  son's  age. 

o 

and  20  X  2  =  40,  the  father's  age. 

4.  A  farmer  has  three  pastures  for  sheep.  In  the  second 
he  has  twice  as  many  as  in  the  first,  and  in  the  third  as 
many  as  in  the  first  and  second  less  ]  5,  and  he  has  in  all 
fifty-seven  :  how  many  has  he  in  each  pasture  1 

5.  What  number  is  that  to  which  .if  ten  be  added,  the 
sum  will  be  equal  to  three  times  the  number  ? 

6.  John  bought  an  equal  number  of  pears,  peaches,  and 
oranges ;  for  which  he  paid  one  dollar  ;  he  paid  a  cent  a  piece 
for  the  pears  and  peaches,  and  three  cents  a  piece  for  the 
oranges  :  how  many  did  he  buy  of  each  sort  1 

7.  A  man  deposited  in  a  savings  bank,  at  different  times, 
eighty  dollars.  The  second  deposit  was  double  the  first,  and 
the  third  was  equal  to  the  first  and  second  and  eight  dollars 
over :  what  was  the  sum  deposited  at  each  time  1 

8.  A  horse,  cart,  and  harness,  together  cost  one  hundred 
and  twenty  dollars  :  the  cost  of  the  horse  plus  twenty  dol- 
lars was  equal  to  the  cost  of  the  cart  and  harness,  and  the 
cart  cost  twenty  dollars  more  than  the  harness. 


26  ELEMENTARY     ALGEBRA. 

LESSON      IX. 

1.  Divide  twenty-one  dollars  between  James,  Jchn,  and 
Charles,  so  that  James  shall  have  four  dollars  more  than 
John,  and  John  one  dollar  more  than  Charles. 

Let  x  =  James'  share  of  the  $21 : 

then         x  —  4  =  John's  share, 
and  x  —  4  —  1  =   Charles'  share, 
and  their  sum,  x  +  x  +  x  —  4  —  4  —  1  =  21: 
hence,  S.r  —  9  =  21. 

Now,  if  ox  diminished  by  9  equals  21,  Sx  must  be  equal 
to  21  increased  by  9, 

therefore,  3x  =  21  +  9  =  30 

or,  x  =  —  =  10. 

o 

Hence,  John's  share  =10  —  4=6, 
and  Charles'  share     =  10  —  5  =  5. 

Eemark. — We  see  from  the  above  example,  that  a  nega- 
tive number  may  be  transposed  from  one  member  of  an 
equation  to  the  other  by  simply  changing  its  sign  to  plus. 

2.  A  person  goes  to  a  tavern  where  he  spends  three  shil- 
lings :  he  then  goes  to  a  second  and  spends  nine  shillings, 
which  is  three  times  as  much  as  he  had  left:  \vhat  had  he  at 
first? 

3.  Three  persons,  A,  B,  and  C,  spend  at  a  tavern,  twenty- 
eight  dollars  :  B  spends  three  dollars  more  than  A,  and  C 
seven  dollars  more  than  B  :  how  much  does  each  spend  1 

4.  There  are  four  numbers  whose  sum  is  33  :  the  second 
is  double  the  first;  the  third  is  three  times  the  second,  and 
the  fourth  is  four  times  the  third  :  what  are  the  numbers? 


INTRODUCTION.  1? 

5.  The  sum  of  two  members  is  13,  and  their  difference 
three  :  what  are  the  numbers  ? 

Let         x  =  the  greater  : 
then    x  —  3  =   the  less  ; 
and   2x  —  3  =  13  :  hence     2x  =  13  +  3  =  16, 

oi  x  =  —  =  8  ;    and    8  —  x  =  5, 

hence,  the  numbers  are  8  and  5. 

6.  James  says  to  John,  "  give  me  five  of  your  marbles 
and  I  shall  have  twice  as  many  as  you  now  have"  :  together 
they  have  nineteen  :  how  many  has  each  % 

Let  x  denote  the  number  which  James  has. 
Then,  19  —  x     will  denote  what  John  has ; 

and  by  the  conditions  of  the  question, 

x  +  5  =  2  (19  —  x)  =  38  —  2a- 
then,  by  transposing  2x  and  5,  we  have 
3a;  =  38  —  5  =  33, 

or,  x  =  —  =11. 

'  3 

Remark. — When  we  wish  to  multiply  an  algebraic  ex- 
pression, composed  of  two  or  more  terms,  by  any  number, 
we  place  those  terms  within  a  parenthesis,  and  write  the 
multiplier  on  the  left,  or  right.     Thus, 

2  (19  -  x)     or     (19  -  r)2 

denotes  that  the  difference  between  19  and  x  is  to  be  multi- 
plied by  2. 

7.  The  sum  of  the  ages  of  a  mother  and  daughter  is  56  : 
the  daughter's  age  is  one-third  of  the  mother's  age :  what  is 
the  age  of  each  ? 


/ 


18  ELEMENTARY      ALGEBRA, 


LESSON      X. 

1.  If  from  3x  we  take  x,  what  will  remain  '?  If  we  takfe 
away  2x,  what  will  be  left  %  If  we  take  away  3x,  what  will 
be  left  % 

2.  If  from  ox,  we  subtract  x  —  1,  what  will  be  left  ? 
Here  we  propose  to  take  from  ox  a  number  less  than  x 

by  1.  If  then,  we  subtract  x  from  3*,  leaving  2.r,  we  shall 
have  taken  too  much,  and  consequently,  the  remainder  will 
be  too  small  by  1.  Hence,  to  obtain  the  true  remainder 
we  must  add  1  ;  and  we  then  have 

2z  —  (z  —  1)  =  3ar  —  *  4-  1  =2s+  1. 
This,  and   all   similar   results,   are    obtained  by  merely 
changing  the  signs  of  the  subtrahend,  and  adding  the  terms. 

3.  What  is  the  difference  between 

4.r  +  3    and    2x  —  2 
4x  +  3  —  (2x  —  2)  =  Ax  —  2x  +  3  -f-  2  =  2x  -f  5. 

4.  What  is  the  difference  between 

Gx  —  9     and     2x  —  8. 

5.  What  is  the  difference  between 

ox  —  4     and     —  x  +  6. 

6.  What  is  the  difference  between 

—  5x  +  7     and     -  3z  +  8. 

7.  James  is  three  years  older  than  John ;  and  one-sixth 
of  James'  age  is  equal  to  one-fifth  of  John's. 

Let  x  denote  James'  age  ; 
then     x  —  3  will  denote  John's ; 
and  by  the  conditions  of  the  question, 


G  5      ' 

hence,  5ar  =  Gx  —  18,     or    x  =  18. 


INTRODUCTION.  19 

8.  William  has  two  cents  more  than  Jo  if  John's 
cents  be  subtracted  from  twice  William's,  the  remain- 
der will  be  ten  :  how  many  has  each  ? 

Let  x  denote  the  number  of  William's ; 
then     x  —  2  will  denote  John's ; 
and  by  the  conditions  of  the  equation, 

2a?  —  (a?  —  2)  =  10  ; 
that  is,        2x  —  x+2   =10 

x  +  2   =  10,  or    x  =  10  —  2  =  8. 

9.  A  farmer  has  sheep  in  two  lots.  In  one  lot  he  has  five 
more  than  in  the  other.  But  three  times  the  larger  flock  is 
equal  to  four  times  the  less :  how  many  are  there  in  each  flock  1 

10.  Lucy  is  five  years  older  than  Jane;  but  four  times 
Lucy's  age,  diminished  by  five  times  Jane's,  is  equal  to 
nothing  :  what  is  the  age  of  each  ? 

11.  What  is  the  difference  between 

5a?  +  3     and    —  7a?  —  4. 

12.  What  is  the  difference  between 

—  6a?  +  3     and         8a;  +  9. 


LESSON    XI. 

1.  A  grocer  buys  an  equal  number  of  lemons  and  orange*  . 
,t>r  the  lemons  he  paid  two  cents  a  piece,  and  for  the  orangoN 
he  paid  three  cents  a  piece,  and  for  the  whole  he  paid  eighty 
cent3  :  how  many  did  he  buy  of  each  sort  1 
Let  x  denote  the  number  of  each  kind  :  then 
2a;  =  the  cost  of  the  lemons, 
and  3a?  =  the  cost  of  the  oranges, 

and  by  the  conditions  of  the  question, 
2z  +  3a;  =  80  cents. 

Heuco,  5a?  =  80 ;    or    x  =  —  =  10. 

o 


20  ELEMENTARY     ALGEBR 

2.  A  grocer  buys  a  certain  number  of  lemons  at  two  cents 
a  piece,  and  three  times  as  many  oranges  at  four  cents  a 
piece,  and  pays  for  the  whole  eighty-four  cents  :  how  many 
does  he  buy  of  each  sorl,  1 

3.  What  number  is  that  which  being  added  to  five  times 
itself,  and  nine  subtracted  from  the  sum,  will  leave  a  re- 
mainder equal  to  21  1 

4.  John  has  in  his  purse  a  certain  number  of  cents,  half 
as  many  dimes  as  cents,  and  half  as  many  dollars  as  dimes  ; — 
in  all  twenty-eight  pieces  :  how  many  has  he  of  each  sort  1 

Let  x  denote  the  number  of  dollars ; 

then  2x  will  denote  the  number  of  dhaes  ; 

and  AlX  the  cents. 

Then,  by  the  conditions  of  the  question, 

28 
x  +  2x  +  4z  =  7x  =  28,    or    x  =  —  =  4. 

5.  In  a  fruit  basket  there  are  three  times  as  many  apples 
as  pears,  and  five  times  as  many  peaches  as  apples  :  in  all, 
ninety-five  :  how  many  of  each  sort  1 

6.  A  farmer  has  sixty-nine  head  of  cattle.     The  number 
of  cows  is  double  that  of  his  calves,  and  the  number  of 
young  cattle  is  six  times  as  great  as  his  calves ;  besides,  he . 
has  six  oxen  :  how  many  calves,  how  many  cows,  and  how 
many  young  cattle  1 

7.  What  number  is  that  which  being  multiplied  by  seven, 
and  five  subtracted  from  the  product,  will  give  a  result  equal 
to  four  times  the  number  increased  by  thirteen  ? 

8.  A  merchant  has  forty-four  dollars  in  bank  bills,  in  an 
equal  number  of  ones,  twos,  threes  and  fives  :  how  many 
has  he  of  each  sort  ? 


INTRODUCTION.  21 

9.  A  jockey  has  a  horse  and  two  saddles,  one  worth 
thirty  dollars  and  the  other  five.  If  he  puts  the  best  saddle 
on  the  horse,  their  value  becomes  double  that  of  the  horse 
diminished  by  twice  the  value  of  the  other  saddle :  what  if 
the  value  of  the  horse  1 


LESSON    XII. 

1.  What  number  is  that  to  which  if  five  be  added,  an 
the  sum  multiplied  by  three,  will  give  a  result  equal  to 
ten  times  the  number  plus  one  ? 

Let  *  denote  the  number. 
Then  by  the  conditions  of  the  question 

3(*  +  5)  =  10*  +  1,  t 

hence,  3*  +  15  =  10*  -f  1  ; 

and  by  transposing  10*  and  15, 

3*  —  10*  =  1  —  15 
or  —  7*    =     —  14, 

and  changing  the  signs  of  both  members, 

14 

7*  =  14    or    *  =  — -  =  2. 

7 

Remark. — When,  after  having  brought  all  the  *'s  to  the 
first  member,  the  final  sign  is  minus,  make  it  plus  by  chang- 
ing the  signs  of  all  the  terms  in  both  the  members. 

2.  The  difference  of  two  numbers  is  three,  and  their  sum 
five  times  the  difference :  what  are  the  numbers? 

3.  James  says  to  John,  "  Give  me  your  apples,  and  I  shall 
then  have  three  times  as  many  as  you  have  now."  John 
says  "  No ;  for  the  number  of  your  apples  now  exceeds 
mine  by  four :"  how  many  had  each  ? 

Let  *  denote  the  number  which  John  had. 

TheD,  ;  +  4  will  denote  what  James  had  ; 


22  ELEMENTARY     ALGEBRA. 

and  by  the  conditions  of  the  question 

x  +  x  -f-  4  =  ox  ;     whence  by- 
transposing,  x  =  4. 

4.  James  met  some  beggars,  to  each  of  whom  he  gave  6 
cents :  had  there  been  four  more,  and  had  he  given  the  same 
to  each,  he  would  have  given  seventy-two  cents  :  how  many 
beggars  were  there  1 

5.  John  has  twice  as  many  turkeys  as  ducks ;  twice  as 
many  ducks  as  geese,  and  eight  times  as  many  chickens  as 
geese ;  in  all,  forty-five :  how  many  has  he  of  each  sort  ? 

6.  Three  persons  receive  forty-eight  dollars  ;  the  second, 
four  dollars  more  than  the  first,  and  the  third,  four  more 
than  the  second  :  how  much  did  each  receive  1 

7.  The  sum  of  three  numbers  is  thirty-six ;  the  second 
exceeds  the  first  by  eight,  and  the  third  is  less  than  the 
second  by  16  :  what  are  the  numbers  ? 

Let  x  denote  the  first  number ; 

then,     x  4-  8  will  denote  the  second  ; 
and  since  the  third  is  less  than  the  second  by  19, 

x  4-  8  —  16  =  x  —  S  =    the  third. 
Then  by  the  conditions  of  the  question 

x4-x  +  %  +  x  —  8  =  36, 
or  3x  =  36,    or   x  =  12. 

Hence,  the  numbers  are  12,  20,  and  4. 

Remark. — When  a  number,  as  -f-  8  and  —  8,  is  found 
twice  in  the  same  member  of  the  equation  and  with  different 
signs,  it  may  be  omitted :  and  the  two  numbers  are  then 
said  to  cancel  each  other.  If  the  same  number  is  found  in 
different  members  of  the  equation  with  the  same  sign,  it 
may  be  omitted :  and  the  two  numbers  are  then  said  to 
cancel  each  :>ther. 


INTRODUCTION.  23 

8.  A  father,  son,  and  daughter,  on  comparing  ages,  find 
that  the  son's  age  is  double  the  daughter's :  that  twice  the 
son's  age  diminished  by  four,  is  equal  to  the  father's  age: 
and  that  the  sum  of  their  ages  is  e^ual  to  73 :  what  is 
the  age  of  each? 


LESSON   XIII. 

1 .  The  sum  of  two  numbers  is  nine  ;  if  to  the  first  six  bo 
added,  the  sum  will  be  double  the  second :  what  are  the 
numbers  1 

Let   x   denote  the  first  number, 
then,      9  —  x    will  denote  the  second; 
and  by  the  conditions  of  the  question 

x  +  C  =  2  (9  —  x)  =  18  —  2x ; 

12 

whence,  Sx  =  12 ;    or  x  =  —  =  4,   the  first, 

o 

and  9  —  x  =  9  —  4  =  5,   the  second  number. 

2.  John  and  James  play  at  marbles :  James,  at  the  be- 
ginning, has  twice  as  many  as  John,  but  John  wins  eight, 
and  he  then  has  twice  as  many  as  James  has  left :  how 
many  had  each,  at  the  beginning  1 

Let    x   denote  John's  marbles, 
then      2x   will  denote  James' ; 
and        x  -f-  8    what  James  had  after  he  won, 
and        2x— 8    denote  what  John  had  after  he  lost. 

Then,  by  the  conditions  of  the  question, 

x  +  8  =  2  (2x— 8)  =  4x  —  16, 
or  3x  =  24  : 

whence,  x  =  8,    the  number  John  had ; 

and  2x  =  16,   the  number  James  had. 


24  ELEMENTARY     ALGEBRA. 

3.  In  an  orchard  containing  sixty  trees,  thei  e  are  twice  as 
many  pear  trees  as  apple  trees,  and  as  many  plum  trees  as 
pear  trees  and  apple  trees  together :  how  many  trees  are 
there  of  each  sort  1 

4.  A  and  B  set  out,  at  the  same  time,  from  two  places 
which  are  ninety  miles  apart,  and  travel  towards  each  other ; 
A  travels  sis  miles  an  hour,  and  B  three  miles  an  hour :  in 
how  many  hours  will  they  meet  % 

Let   x  denote  the  number  of  hours : 
then      Qx   will  denote  the  number  of  miles  A  travels, 
and        Zx   the  number  of  miles  B  travels  : 
By  the  conditions  of  the  question 

6x  -f  ox  =  90, 
whence  9x  =z  90   or  x  =  10. 

5.  Charles  buys  six  yards  of  cloth  at  a  certain  price,  and 
afterwards  nine  yards  more  at  the  same  price,  but  the  last 
time  he  paid  twenty -seven  shillings  more  than  before :  how 
much  did  he  pay  a  yard  1 

6.  A  cask  which  holds  eighty  gallons  is  filled  with  a  mix- 
ture of  brandy,  wine,  and  cider :  there  are  ten  gallons  more 
of  cider  than  of  wine,  and  as  much  brandy  as  of  cider  and 
wine  together  :    how  many  gallons  are  there  of  each  ? 

7.  Four  men  build  a  boat  together,  which  cost  one  hun- 
dred and  twenty-one  dollars  :  the  second  paid  twice  as  much 
as  the  first,  the  third  as  much  as  the  first  and  second  together, 
and  the  fourth  as  much  as  the  third  and  second  together  : 
what  did  each  pay? 

8.  B  has  six  shillings  more  than  A ;  C  has  six  shillings 
more  than  B  ;  D  has  six  shillings  more  than  C ;  D  has  also 
three  times  as  many  shillings  as  A  :  how  many  shillings  has 
each? 


INTRODUCTION.  25 

9.  At  an  election  one  hundred  votes  were  given,  and  the 
successful  candidate  had  a  majority  of  twenty :  how  many 
votes  had  each  candidate  % 

10.  Two  men  together  had  twenty  dollars ;  and  they 
played  till  one  lost  five  dollars,  when  the'  winner  had  four 
times  as  much  as  the  loser :  how  much  had  each  when 
they  began  % 

11.  Divide  fifteen  into  two  parts,  such  that  one  part  shall 
be  equal  to  twice  the  other. 

12.  A  fish  was  caught  which  weighed  twenty  pounds ;  the 
head  weighed  four  times  as  much  as  the  tail,  and  the  body 
weighed  five  times  as  much  as  the  tail  %  What  did  each 
part  weigh  1 


LESSON   XIV. 

1.  John  has  a  certain  number  of  marbles:  Charles  has 
half  as  maDy,  and  James  one-third  as  many :  together  they 
have  eleven :  how  many  has  each  ? 

Let    x  =    the  number  of  John's  marbles ; 
then,     —  =    the  number  which  Charles  has, 

At 

X 

and       —  =    the  number  which  James  has. 
o 

Then,  by  the  conditions  of  the  question, 

x         x 

To  clear  the  equation  of  fractions,  multiply  each  member 
by  the  least    common    multiple  of  the  denominators  (see 
Art.  68),  which  in  this  case  is  6,  and  we  shall  have 
2 


26  ELEMENTARY  ALGEBRA, 

Gx  +  Sz  +  2x   =  66, 

hence,  11a;  =  66   or   x  =  —  =  6. 

XX  X 

2.  What  is  the  sum  of  — ,      —      and      -—  ? 

4>  o  4 


SOLUTION 

l  +  l  +  l-A-fl  +  A-i!-!! 

2   T  3    T  4         12  ^  12  ^  12  ~  12  ~    "' 
Hence,  the  sum  is  lyg£. 

8.  What  number  is  that,  which  being  added  to  half  itself 
to  one-third  of  itself,  and  to  one-fourth  of  itself,  will  give  a 
sum  equal  to  twenty -five? 

4.  What  number  added  to  its  fifth  part,  will  give  a  result 
equal  to  twice  the  number  diminished  by  eight? 

5.  What  number  is  that  to  which,  if  three-fourths  of  it 
self  be  added,  the  sum  will  be  twice  the  number  diminished 
by  two  ? 

6.  A  farmer  has  twice  as  many  oxen  as  horses ;  and  one- 
third  the  number  of  his  horses,  added  to  half  the  number 
of  his  oxen,  is  equal  to  four :  how  many  oxen  and  horses 
had  he  ? 

7.  James  has  fifteen  oranges,  which  are  three-fourths  as 
many,  as  John  has,  less  three  :  how  many  has  John  1 

8.  The  smaller  of  two  numbers  is  five-eighths  of  the 
larger,  and  their  sum  is  sixty -five  :  what  are  the  numbers  1 

9.  The  smaller  of  two  numbers  is  three-fourths  the 
larger,  and  their  difference  is  equal  to  half  the  greater 
diminished  by  two :  what  are  the  numbers  1 

10.  A  farmer  sold  a  cow  and  calf:  he  received  one-fifth 
as  much  for  the  calf  as  for  the  cow ;  and  the  difference  be- 


INTRODCCTION.  27 

tween  the  two  sums  was  twenty  four  dollars :  what  did  he 
receive  for  each  1 

11.  James  is  six  years  older  than  John;  and  the  sum  of 
their  ages  plus  one-fourth  of  John's  age,  is  equal  to  twenty- 
four  :  what  is  the  age  of  each  1 

12.  Nancy's  age  is  three  times  Eliza's :  one-half  Nancy's 
plus  one-third  of  Eliza's,  is  equal  to.  the  difference  of  their 
Bges  diminished  by  one :  what  is  the  age  of  each  ? 

IS.  John  is  nine  years  older  than  his  sister.  If  one-sixth 
of  his  age  be  added  to  his  sister's,  the  sum  will  be  two-thirds 
of  John's.     What  was  the  age  of  each  ? 

14.  The  difference  between  two  numbers  is  four;  and 
one-third  of  the  less  plus  one-fourth  the  greater  is  equal  to 
half  the  greater  :  what  are  the  numbers  % 

15.  A  pole  is  one- third  in  the  mud,  one-half  in  the  water, 
and  six  feet  out  of  water  :  what  is  the  length  of  the  pole  1 

16.  The  weight  of  a  fish  is  thirty-two  pounds  :  one-third 
the  weight  of  his  head  is  equal  to  the  weight  of  his  tail ; 
and  the  weight  of  his  body  is  four  times  the  weight  of  his 
tail :  what  is   the  weight  of  each  part  1 

17.  A  person  in  play  lost  one-fourth  of  his  money,  when 
he  found  that  he  had  one-half  of  what  he  began  with  and 
five  shillings  over :  how  much  had  he  when  he  began  to 
play? 


LESSON   XV. 

1.  The  sum  of  the  ages  of  Jane  and  Catharine  is  eigh- 
teen :  but  one-half  of  Catharine's  age  is  equal  to  one-fourth 
of  Jano's  age :  what  is  the  age  of  esieh  1 


28  ELEMENTARY     ALGEBRA. 

Let   x   denote  Jane's  age  ; 
then       18  —  x   will  denote  Catharine's  age; 
and  by  the  conditions  of  the  question 

18  —  x       x 

~~2      ~T' 

Multiplying  each  member  of  the  equation  by  four,  we 
have 

36  —  2x  =  x\    whence    36  =  Sx 

or  Sx  =  36,    or   z  =  — ■  =  12. 

o 

2.  If  from  one-fifth  of  a  man's  money,  one-sixth  be  taken, 
he  will  have  one  dollar  left :   how  much  has  he  1 

Let   x   denote  the  amount  which  he  has  : 
Then  by  the  conditions  of  the  question 
x         x 
~5        ~6~  ~ 
Multiplying  both  members  of  the   equation  by  30,  the 
least  common  multiple  of  the  denominators,  and  we  have 

Qx  —  hx  =  30,     whence,     x  =  30. 

3.  John  sells  one-third  of  his  eggs,  and  then  one-half  of 
what  he  first  had,  after  which  he  has  three  left :  how  many 
had  he  at  first? 

4.  John  gave  one-third  of  his  apples  to  Charles,  and 
Charles  gave  one-fourth  of  what  he  received  to  William, 
and  then  had  six  left :  how  many  apples  had  John  % 

a  certain  number  be  diminished  by  three,  and  one- 
enird  of  the  remainder  be  subtracted  from  the  number,  the 
result  will  be  equal  to  eleven :  what  is  the  number  1 

6.  The  difference  between  five-sixths  of  a  number  and  one- 
third  of  the  same,  number  is  nine     what  is  the  number? 


IITTRODUCTIOH.  29 

7.  The  difference  between  four-fifths  of  a  number  and  one- 
third  of  a  number  is  seven  :  what  is  the  number  ? 

8.  If  from  five-eighths  of  a  number  we  take  one  half  the 
number,  and  then  take  one  from  the  difference,  the  result  will 
be  equal  to  nothing  :  what  is  the  number  1 

9.  A  market  woman  bought  a  certain  number  of  eggs, 
one-third  of  which  she  sold  :  five  of  the  eggs  spoiled,  and 
she  then  had  just  three-quarters  of  a  dozen  left :  how  many 
did  she  buy  ? 

10.  The  difference  between  one-half  of  a  number  and  one- 
fifth  of  it  is  three  :  what  is  the  number  1 

11.  What  is  the  value  of  x,  in  the  equation 

x        x        ~        x 

T  +  ¥  +  3  =  -6+6- 

12.  What  is  the  value  of  x,  in  the  equation 

0  it       '  <& 

13.  What  is  the  value  of  x,  in  the  equation 

x        x 

—  —  —  +  2  =5  —  z. 


LESSON    XVI. 

1 .  If  a  laborer  can  do  a  piece  of  work  in  five  days,  what 
part  of  it  can  he  do  in  one  day  1 

2.  If  James  can  do  a  piece  of  work  in  eight  days,  how 
much  of  it  can  he  do  in  one  day  ?  How  much  in  two  days  1 
How  much  in  three  days  1     How  much  in  x  days  ' 

3.  James  can  do  a  piece  of  work  in  three  days,  and  John 
can  do  it  in  six  days :  in  how  many  days  can  they  both  do 
it,  workirg  togother  1 


30  ELBMESTiRT    ALGEBRA, 

Let       1   denote  the  work  to  be  done  ; 
and  x  =z  the  time  in  which  both  can  do  it,  together. 

Then,      —  =  what  James  can  do  in  one  day, 

o 

x 
end  —  =  what  James  can  do  in  z  days  : 

o 

also,        —  =  what  John  can  do  in  one  day, 

x 

and  —  =  what  John  can  do  in  £  days, 

X  X 

Then,      — -  +  — -  =  1,  the  work  done ; 
3         o 

and  by  clearing  the  equation  of  fractions, 

2x  +  x  =  0,    or     Sx  =  6 ; 

hence,        x  =  -5-  =  2. 
o 

Therefore,  together,  they  can  do  the  work  in  two  days. 

4.  If  A  can  do  a  piece  of  work  in  four  days,  and  B  can 
do  the  same  work  in  twelve  days,  how  long  will  it  take 
both  of  them  to  do  the  same  work  % 

Let  x  denote  the  time  :  then  by  the  conditions  of  the 
question, 

x        x 

T 

whence,  x  is  found  equal  to  3, 

5.  If  Charles  can  do  a  piece  of  work  in  five  days,  and 
John  ill  twenty  days  :  how  long  will  it  take  both  of  them, 
working  together,  to  do  the  same  work  1 

6.  A  barrel  can  be  emptied  by  one  faucet  in  six  hours, 
and  by  another  in  thirty  hours  :  how  long  will  it  take  both 
to  empty  it,  running  together  ? 


f12==1; 


IKTftODUCTlOST.  31 

If.  A  hogshead  can  be  emptied  by  one  faucet  in  seven 
hours,  and  by  another  in  forty-two  hours  :  how  long  will  it 
take  both  to  empty  it,  running  together  1 

8.  If  A  can  do  a  piece  of  work  in  four  days,  B  in  five 
days,  and  C  in  six  days ;  in  how  many  days  will  they  per- 
form  it,  when  working  together  ? 

Let       1  denote  the  work  to  be  done  ; 
and  x  denote  the  number  of  days. 

1 

Then,       —  =  what  A  can  do  in  one  day  ; 

x 
and  —  =   what  he  can  do  in  x  days ; 

—  =  what  B  can  do  in  one  day, 

End  —  =  what  he  can  do  in  x  days. 

1 

Also,       —  =  -what  C  can  do  in  one  day, 

x 
and  —  =  what  C  can  do  in  x  days. 

G  J 

Then,  by  the  conditions  of  the  question, 

4  +   5  +  6  ' 

the  multiplying  by  60,  the  least  common  multiple  of  the 
denominators,  we  have 

15*  -f  lSte+  lOz  =  60; 

whence,  <c  =  —    =  Iff  days« 

9.  An  orchard  of  pear  and  cherry  trees  has  twenty  trees 
of  both  sorts  :  if  the  number  of  pear  trees  be  diminished 
by  twice  the  number  of  cherry  trees,  the  remainder  will  be 
equal  to  5  :  how  many  are  there  of  each  sort  1 


32  ELEMENTARY     ALGEBRA. 

10.  James  bought  a  pencil  and  a  knife.,  for  which  he  paid 
one  dollar :  what  he  paid  for  the  pencil,  diminished  by  twice 
what  he  paid  for  the  knife,  is  equal  to  minus  twenty  :  what 
did  he  pay  for  each  ] 

11.  Divide  twenty-four  into  two  such  parts  that  the 
greater  diminished  by  twice  the  less  shall  be  equal  to  the 
loss  :  what  are  the  parts  ? 

12.  James  and  John  together  have  twenty  oranges.  Four 
times  John's  oranges  taken  from  twice  James',  leaves  a 
remainder  equal  to  half  the  whole  number  of  oranges:  how 
many  had  each1? 

13.  James  buys  a  number  of  oranges.  He  gives  three 
away,  and  then  divides  the  remainder  equally  among  eight 
boys.  Now,  the  whole  number  of  oranges  diminished  by 
the  share  of  each  boy,  is  equal  to  seventeen  :  how  many 
oranges  did  he  buy  1 

14.  The  difference  between  a  father's  age  and  his  son' 
age  is  24  years.     But  if  the  father's  age  be  diminished  by 
twice  the  son's  age,  the  remainder  will  be  four :  what  is  the 
age  of  the  father  % 

15.  A  drover  sold  one-third  of  his  cattle  to  one  man,  and 
one-third  of  the  remainder  to  another,  and  then  had  sixteen 
left :  how  many  had  he  at  first  1 

16.  A  man  goes  to  a  tavern,  where  he  spends  three  shil- 
lings :  he  then  borrows  as  much  as  he  has  left,  and  finds 
that  the  amount  in  his  purse  is  more  than  what  he  had  at  fi'st 
by  four  shillings  :  how  much  had  he  at  first  ? 


ELEMENTARY    ALGEBRA. 


CHAPTER  1. 

Preliminary  Definitions  and  Remarks. 

1.  Quantity  is  a  general  term  applied  to  every  thing 
which  can  be  increased  or  diminished,  estimated  or  measured. 

2.  Mathematics  is  the  science  of  quantity. 

3.  Algebra  is  that  branch  of  mathematics  in  which  the 
quantities  considered  are  represented  by  letters,  and  the 
operations  to  be  performed  upon  them  are  indicated  by 
signs.     These  letters  and  signs  are  called  symbols. 

4.  The  sign  +,  is  called  phis  ;  and  indicates  the  addition 
of  two  or  more  quantities.  Thus,  9  -f  5,  is  read,  9  plus  5, 
or  9  augmented  by  5. 

If  we  represent  the  number  nine,  by  the  letter  a,  and 
the  number  5  by  the  letter  b,  we  shall  have  a  +  b,  which  is 
read,  a  plus  b  ;  and  denotes  that  the  number  represented  by 
a  is  to  be  added  to  the  number  represented  by  b. 

5.  The  sign  — ,  is  called  minus ;   and  indicates  that  one 

1.  What  is  quantity  ? 

2.  What  is  Mathematics  ? 

3.  What  is  Algebra  ?     What  are  the  letters  and  signs  called  I 

4.  What  does  the  sign  plus  indicate  ? 

5.  What  does  the  sign  minus  indicate  ? 

2* 


34  ELEMENTARY     ALGEBRA. 

quantity  is  to  be  subtracted  from  another.     Thus,  9  —  5  is 
read,  9  minus  5,  or  9  diminished  by  5. 

In  like  manner,  a  —  b,  is  read,  a  minus  b,  or  a  diminished 
by  b. 

6.  The  sign  X ,  is  called  the  sign  of  multiplication  ;  and 
tfhen  placed  between  two  quantities,  it  denotes  that  they 
&re  to  be  multiplied  together.  The  multiplication  of  two 
quantities  is  also  frequently  indicated  by  simply  placing  a 
point  between  them.  Thus,  36  X  25,  or  36.25,  is  read,  36 
multiplied  by  25,  or  the  product  of  36  by  25. 

7.  The  multiplication  of  quantities,  which  are  represented 
by  letters,  is  indicated  by  simply  writing  the  letters  one  after 
the  other,  without  interposing  any  sign. 

Thus  ab  signifies  the  same  thing  as  a  X  b,  or  as  a.b  \ 
and  abc  the  same  as  a  X  b  X  c,  or  as  a.b.c.  Thus,  if  we 
suppose  a  =  36,  and  b  =  25,  we  have 

ab  =  36  X  25  =  900. 

Again,  if  we  suppose  a  =  2,  b  =  3  and  c  =  4,  we  have 

abc  =  2  X  3  X  4  =  24. 

It  is  most  convenient  to  arrange  the  letters  of  a  product 
in  alphabetical  order. 

8.  In  a  product  denoted  by  several  letters,  as  abc,  the 
single  letters,  a,  b,  and  c,  are  called  literal  factors  of  the 
product.  Thus,  in  the  product  ab,  there  are  two  literal  fac- 
tors, a  and  b  ;  in  the  product  abc,  there  are  three,  a,  b,  and  c. 

0.  "What  is  the  sign  of  multiplication  ?  "What  does  the  sign  of  multi- 
plication indicate  ?     In  how  many  ways  may  multiplication  be  expressed! 

7.  If  letters  only  are  used,  how  may  their  multiplication  be  expressed 

8.  In  the  product  of  several  letters,  what  is  each  letter  called  ?  How 
"iany  factors  in  ab? — In  abc? — In  abed? — In  abedf? 


DEFINITION     OF     TERMS.  35 

9.  There  are  three  signs  used  to  denote  division.     Thus, 

fl-ri  denotes  that  a  is  to  be  divided  by  b. 

a 

7  denotes  that  a  is  to  be  divided  by  b. 

b 

a  |  b      denotes  that  a  is  to  be  divided  by  b. 

10.  The  sign  =  ,  is  called  the  sign  of  equality,  and  is 
read,  is  equal  to.  W  hen  placed  between  two  quantities,  it 
denotes  that  they  are  equal  to  each  other.  Thus,  9  —  5  =  4 : 
that  is,  9  minus  5  is  equal  to  4 :  Also,  a  +  b  =  c,  denotes 
that  the  sum  of  the  quantities  a  and  b  is  equal  to  c. 

If  we  suppose  a  =  10,  and  b  -—  5,  we  have 

a  -\-  b  —  c,     and     10  +  5  =  c  =  15. 

11.  The  sign  >,  is  called  the  sign  of  inequality,  and  is 
used  to  express  that  one  quantity  is  greater  or  less  than 
another. 

Thus,  a  >  b  is  read,  a  greater  than  b  ;  and  c  <  d  is 
read,  c  less  than  d;  that  is,  the  opening  of  the  sign  is 
turned  towards  the  greater  quantity.  Thus,  if  a  =  9,  and 
6  =  4,  we  write,  9  >  4. 

12.  If  a  quantity  is  added  to  itself  several  times,  as 
a-i-a-j-a  +  a  +  «  +  a,  we  generally  write  it  but  once,  and 
then  place  a  number  before  it  to  show  how  many  times  it 
is  taken.     Thus, 

a-\-a-{-a-\-a-\-a=  5a. 

9,  How  many  signs  are  used  in  division  ?     What  are  they  ? 

10.  What  is  the  sign  of  equality  ?  When  placed  between  two 
quantities,  what  does  it  indicate  ? 

Hi  For  what  is  the  sign  of  inequality  used?  Which  quantity  is 
placed  on  the  side  of  the  opening  ? 

12.  What  is  a  co-efficient  ?  How  many  times  is  ab  taken  in  the  ex- 
pression abl  InSai?  In4a&?  In6a6?  In  &ab ?  If  no  coefficient 
is  written,  what  co-efficient  is  understood  ? 


86  ELEMENTARY     ALGEBKA. 

The  number  5  is  called  the  co-efficient  of  a,  and  denotes 
that  a  is  taken  5  times. 

If  the  co-efficient  is  1,  it  is  -generally  omitted.  Thus,  a 
and  1  a  are  the  same,  each  being  equal  to  a,  or  to  one  a. 

13.  If  a  quantity  be  multiplied  continually  by  itself,  as 
a  X  «  X  a  X  a  X  «,  we  generally  express  the  product  by 
writing  the  letter  once,  and  placing  a  number  to  the  right 
of,  and  a  little  above  it :   thus, 

a  X  «  X  a  X  a  X  «  =  «5- 

The  number  5  is  called  the  exponent  of  a,  and  denotes 
the  number  of  times  which  a  enters  into  the  product,  as  a 
factor.  For  example,  if  we  have  a3,  and  suppose  a  =  3, 
we  write, 

a?  —  ax  a    X  a   =  33  =3x3x3  =  27. 
If  a  =  4,  a3  =  43  =  4   X  4  X  4  =  G4, 

and  for  a  =  5,       a3  =  53  =  5    X  5  X  5  =  125. 

If  the  exponent  is  1,  it  is  generally  omitted.  Thus,  a1  is 
the  same  as  «,  each  expressing  that  a  enters  but  once  as  a 
factor. 

14.  The  power  of  a  quantity  is  the  product  which  results 
from  multiplying  that  quantity  by  itself  a  certain  number 
of  times.     Thus, 

a3  =  43 -  =  4  X  4  X  4  =  G4, 
64  is  the  third  power  of  4,  and  the  exponent  3  shows  the 
degree  of  the  power. 

15.  The  sign  y/     ,  is  called  the  radical  sign,  and  when 

13.  What  does  the  exponent  of  a  letter  denote  ?  How  many  times 
is  a  factor  in  as  ?  In  a3  ?  In  a*  ?  In  a6  ?  If  no  exponent  is  written, 
what  exponent  is  understood  ? 

14.  What  is  the  power  of  a  quantity  ?  What  is  the  third  power 
of  2  ?     Express  the  fourth  power  of  a  ? 

15.  Express  the  square  root  of  a  quantity?  Also  the  cube  root 
Aloo  the  4th  root 


DEFINITION     OF     TERMS.  37 

prefixed  to  a  quantity,  indicates  that  its  rout  is  u      e  ex- 
tracted.    Thus, 

ya  or  simply  -y/a  denotes  the  square  root  of  a. 
•(/^"denotes  the  cube  root  of  a. 
^/a~denotes  the  fourth  root  of  a. 
The  number  placed  over  the  radical  sign,  is  called  the  in- 
dex of  the  root.      Thus,  2  is  the  index  of  the  square  root,  3 
of  the  cube  root,  4  of  the  fourth  root,  &c. 
If  we  suppose  a  =  64,  we  have 

y/64  =:  8,  .J/64  =  4. 

16.  Every  quantity  written  in  algebraic  language,  that 
is,  with  the  aid  of  letters  and  signs,  is  called  an  algebraic 
quantity,  or  the  alegebraic  ex2)ression  of  a  quantity.    Thus, 
(  is    the    algebraic    expression   of    three 
I       times  the  number  a  ; 
is  the  algebraic  expression  of  five  times 
the  square  of  a  ; 
(  is   the    algebraic    expression   of   seven 
7a362  \       times  the  product  of  the  cube  of  a  by 
(       the  square  of  b ; 

(  is  the  algebraic  expression  of  the  difFer- 
3a  —  56  <       ence  between  three  times  a  and  five 
(       times  b ; 

'is  the  algebraic  expression  of  twice  the 
square  of  a,  diminished  by  three  times 
the  product  of  a  by  5,  augmented  by 
four  times  the  square  of  b. 
]     Write  three  times  the  square  of  a  multiplied  by  the 
cube  of  b.  Ans.  3a2b3 

16.  What  i9  an  algebraic  quantity  ?      la  5ab  an  algebraic  quantity! 
Js  9o       Is  4v  1     Is  36  —  x  ?     Give  other  examples. 


5a2 


2fct2  —  Sab  +  lb2  < 


38  ELEMENTAL Y     ALGEBRA. 

2.  Write  nine  times  the  cube  of  a  multiplied  by  b,  dimin 
ished  by  the  square  of  c  multiplied  by  d.      Ans.  9a36  —  c2d 

3.  If  a  =  2,  b  =  3,  and  c  =  5,  what  will  be  the  value  of 
3a2  multiplied  by  62,  diminished  by  a  multiplied  by  b  mul 
tiplied  by  c.     We  have 

3a262  -  abc  =  3  x  22  x  32  -  2  X  3  x  5  =  78. 

4.  If  a  z=  4,  6=6,  c  =  7,  d  =  8,  what  is  the  value  of 
9a2  +  6c  -  ad?  Ans.  154. 

5.  If  a  =  7,  6  =  3,  c  =  7,  a7  =  1,  what  is  the  value  of 
Gao7  +  362c  —  4cZ2  ]  ylns.  227 

6.  If  a  =  5,  6  =  6,  c  :=  6,  d  =  5,  what  is  the  value  of 
9abc  —  Sad  +  46c  1  Ans.  1564. 

7.  Write  ten  times  the  square  of  a  into  the  cube  of  b  into 
c  square  into  the  cube  of  d. 

17.  When  an  algebraic  quantity  is  not  connected  with 
any  other,  by  the  sign  of  addition  or  subtraction,  it  is  called 
a  monomial,  or  a  quantity  composed  of  a  single  term,  or  sim- 
ply, a  term.     Thus, 

3a,     5a2,     7a362, 
are  monomials,  or  single  terms. 

18.  An  algebraic  expression  composed  of  two  or  more 
parts,  connected  by  the  sign  +  or  — ,  is  called  a  polynomial, 
or  quantity  composed  of  two  or  more  terms.    For  example, 

3a  —  56     and     2a2  —  3c6  +  462 
are  polynomials. 

19.  A  polynomial  composed  of  two  terms,  is  called  a 
binomial  ;  and  one  of  three  terms,  is  called  a  trinomial. 

17.  TVhat  is  a  monomial  ?     Is  Sab  a  monomial  ? 

18.  What  is  a  polynomial  ?     Is  Za  —  6  a  polynomial  ? 

19.  "Wliat  is  a  binomial  ?     Wbat  is  a  triuomial  I 


DEFINITION     OF     TERMS.  39 

20.  Each  of  the  literal  factors  which  compose  a  term  is 
called  a  dimension  of  the  term :  and  the  degree  of  a  term  is 
the  number  of  these  factors  or  dimensions.     Thus, 

j  is  a  term  of  one  dimension,  or  of  the 

(      first  degree. 

(  is  a  term  of  two  dimensions,  or  of  the 

(      second  degree. 

_,  „  .     w       ,       I  is  of  six  dimensions,  or  of  the  sixth  de- 
late2 =laaabcc  i 

{      gree. 

21.  A  polynomial  is  said  to  be  homogeneous,  when  all  its 
terms  are  of  the  same  degree.     Thus,  the  polynomial 

3a  —  26  +  c    is  of  the  first  degree,  and  homogeneous. 
—  4ab  +  b2    is  of  the  second  degree,  and  homogeneous. 
5a2c  —  4c3  +  2c2d    is  of  the  third  degree,  and  homogeneous. 
8a3  +  4ab  +  c    is  not  homogeneous. 

22.  A  vinculum,  or  bar ,  or  a  parenthesis  (  ), 

is  used  to  express  that  all  the  terms  of  a  polynomial  are  to 
be  considered  together.     Thus, 

a  4-  b  +  c  X  b,    or    (a  -\-  b  +  c)  X  b, 
denotes,  that  the  trinomial  a  +  b  +  c,  is  to  be  multiplied  by  6; 
also,  a  +  6  +  c  X  c-\-  d  -\-f,  or  (a  +  b  +  c)  X  (c  +  d  +/), 
denotes  that  the  trinomial  a  +  b  +  c,  is  to  be  multiplied  by 
the  trinomial  c  +  d  +/• 

When  the  parenthesis  is  used,  the  sign  of  multiplication 
is  usually  omitted.     Thus, 

(a  -{-  b  -\-  c)  x  b    is  the  same  as    (a  +  6  +  c)6. 

20.  What  is  the  dimension  of  a  term  ?  What  is  the  degree  of  a 
term  ?  How  many  factors  in  Zabc  ?  Which  are  they  ?  What  is  its 
degree  ? 

21.  When  is  a  polynomial  homogeneous?  Is  the  polynomial 
Za?b  +  3a=62  homogeneous  ?     Is  2a46  —  b3  ? 

22i  For  what  is  the  vinculum  or  bar  used  ?  Can  you  express  the 
same  with  the  parenthesis  ? 


40  ELEMENTARY      ALGEBRA. 

23.  If  two  or  more  terms  of  a  polynomial  contain  the 
same  letters,  and  the  same  letter  in  each  have  the  same  ex 
ponent,  such  are  called  similar  terms. 

Thus,  ha  the  polynomial 

lab  +  Sab  -  4a3b2  4-  5a352, 
the  terms  lab,  and  3ab,  are  similar :  and  so  also  are  the 
terms  —  A.a?b2  and  5a3b2,  the  letters  and  exponents  in  both 
being  the  same.  But  in  the  binomial  &a2b  -f-  lab2,  the 
terms  are  not  similar ;  for,  although  they  are  composed  of 
the  same  letters,  yet  the  same  letter  in  each  is  not  affected 
with  the  same  exponent. 

REDUCTION    OF   ALGEBRAIC    EXPRESSIONS. 

24.  The  simplest  form  of  a  polynomial,  is  an  equivalent 
expression  containing  the  fewest  terms  to  which  it  can  be 
reduced.  When  a  polynomial  contains  similar  terms,  it 
may  be  reduced  to  a  simpler  form. 

1.  Thus,  the  expression  Sab  -f-  2a6,  is  evidently  equal 
to  5ab. 

2.  Reduce  the  polynomial  Sac  4-  9ac  4-  2ac  to  its  sim- 
plest form.  Ans.  14ac. 

3.  Reduce  the  polynomial  abc  4-  4a£c  +  babe  to  its  sim- 
plest form. 

In  adding  similar  terms  together  we  abc 

take  the  sum  of  the  co-efficients  and  4abc 

annex  the  literal  part.    The  first  term,  5abc 

abc,   has    a    co-efficient  1   understood,  lOabc 
(Art.  12). 


23.  What  are  similar  terms  of  a  polynomial  ?  Are  3asi  and  6aJ^s 
similar  ?     Are  2a263  and  2a363  ? 

24.  What  is  the  simplest  form  of  a  polynomial  ?  If  the  terms  are 
positive  and  similar,  may  they  be  reduced  to  a  simpler  form  ?  In  what 
way? 


DEFINITION      OF     TERMS.  41 

25.  Of  the  different  terms  -which  compose  a  polynomial, 
iome  are  preceded  by  the  sign  -{->  and  the  others  by  the 
sign  — .  The  former  are  called  additive  terms,  the  latter, 
sub  tractive  terms. 

When  the  first  term  of  a  polynomial  is  not  preceded  by 
any  sign,  it  is  understood  to  be  affected  with  the  sign  -f. 

1.  John  has  20  apples  and  gives  5  to  William:  how 
many  has  he  left1? 

Now,  let  us  represent  the  number  of  apples  which  John 
has  by  a,  and  the  number  given  away  by  b :  the  number  he 
has  left  will  then  be  represented  by  a  —  b. 

2.  A  merchant  goes  into  trade  with  a  certain  sum  of 
money,  say  a  dollars ;  at  the  end  of  a  certain  time  he  has 
gained  b  dollars :  how  much  will  he  then  have  1 

If  instead  of  gaining,  he  had  lost  b  dollars,  how  much 
would  he  have  had  1  Ans.  a  —  b  dollars. 

Now,  if  the  losses  exceed  the  amount  with  which  he 
began  business,  that  is,  if  b  were  greater  than  a,  we  must 
prefix  the  minus  sign  to  the  remainder  to  show  that  the 
quantity  to  be  subtracted  was  the  greatest. 

Thus,  if  he  commenced  business  with  $2000,  and  lost 
$3000,  the  true  difference  would  be  —  $1000:  that  is,  the 
subtractive  quantity  exceeds  the  additive  by  $1000. 

3.  Let  a  merchant  call  the  debts  due  him  additive,  and 
the  debts  he  owes,  subtractive.  Now,  if  he  has  due  him 
$600  from  one  man,  $800  dollars  from  another,  $300  from 
another,  and  owes  $500  to  one,  $200  to  a  second,  and  $50 
to  a  third,  how  will  the  account  stand  1      Ans.  $950  due  him. 

25.  What  are  the  terms  called  which  are  preceded  by  the  sign  +  ? 
What  are  the  terms  called  which  are  preceded  by  the  sign  —  ?  If  no 
aign  is  prefixed  to  a  term,  what  sign  is  understood !  If  some  of  the 
terms  are  additive  and  some  subtractive,  may  they  be  reduced  if  simi- 
lar? Give  the  rule  for  reducing  them.  Does  the  reduction  alfcct  the 
uxpoueiiis,  or  only  fhe  co-efficients? 

3 


42  E  L  L,  M  E  N  T  A  11  Y      A  L  U  E  B  R  A  . 

4.  Reduce  to  its  simplest  form  the  expression 

2a2b  +  5a2b  —  3a26  +  A.a2b  -  Qa2b  —  a2b. 
Additive  terms,  Subtractive  terms. 


+    oa2b 

-    3a2b 

+    5a25 

-    Ga2b 

-f    4a25 

-      a2b 

Sum  +  12a26 

Sum 

—  10a26. 

it,                      12a25  - 

10«26  = 

=  2a2b. 

Hence,  for  the  reduction  of  the  similar  terms  of  a  polyno- 
mial we  have  the  following 

RULE. 

I.  Add  together  the  co-efficients  of  all  the  additive  terms, 
and  annex  to  tJieir  sum  the  literal  part ;  and  form  a  single 
subtractive  term  in  a  similar  manner. 

II.  Then,  subtract  the  less  co-efficient  from  the  greater, 
and  to  the  remainder  prefix  the  sign  of  the  greater  co- 
efficient, to  which  annex  the  liter cd  part. 

Remark. — It  should  be  observed  that  the  reduction  affects 
only  co-efficients,  and  not  the  exponents. 

EXAMPLES. 

1.  Reduce  to  its  simplest  form  the  polynomial 
+  2a3bc2  -  4a3bc2  +  6a3bc2  -  $a3bc2  +  \\a3bc\ 
Find  the  sum  of  the  additive  ard  subtractive  terms  sepiv 
rately,  and  take  their  difference :  thus, 

Additive  terms.  Subtractive  ter?ns. 

+    2a3bc2  —   4a3bc2 

-f    6a3bc2  —   8a3bc2 

+  Ua3bc2  Sum    —  ~12^36c" 

Sum  +  I9a3bc2 

Hence,  we  have,  I9a3bc2  —  VZa3bc2  —  7a3bc2. 


ADDITION,  43 

2.  Reduce  the  polynomial  4a2b  —  8a2b  —  9a2b  -f  1 1  az& 
to  its  simplest  form.  Ans.   —  2a26. 

3.  Reduce  the  polynomial  7abc2  —  abc2  —  labc2  4-  8a6c2 
-f  6aic2   to  its  simplest  form.  Ans.       loabc2. 

4.  Reduce  the  polynomial  9cb3  —  Sac2  +  15c63  +  8ca 
-f  9ac2  —  24c63    to  its  simplest  form.  Ans.  ac2  -(-  8ca. 

The  reduction  of  similar  terms  is  an  operation  peculiar  to 
algebra.  Such  reductions  are  constantly  made  in  Algebraic 
Addition,  Subtraction,  Multiplication,  and  Division. 


ADDITION. 

26.  Addition  in  Algebra,  is  the  process  of  finding  the 
simplest  equivalent  expression  for  several  algebraic  quan 
tities.     Such  equivalent  expression  is  called  their  sum. 

1.  What  is  the  sum  of 

Sax  +  2ab   and    +  2a#  +  ab. 

3ax  -f-  2ab 

We  reduce  the  terms  as  in  Art.  25,        —  2ax  -f-    ab 

and  find  for  the  sum         ax  -f-  oab 

{*> 
2c 


The  result  is 3a  +  56  +  2c 

an  expression  which  cannot  be  reduced  to  a  more  simple 
form. 

26 1  What  is  addition  in  Algebra  ?      What  iss  such  simplest  and  equi- 
Talent  expression  called  I 


4a263 

2a263 

7a263 

13a263 

2a2 

-4ab 

3a2 

—  3a6-f 

62 

2a&- 

562 

5a2 

—  oab  — 

462 

44  ELEMENTARY     ALGEBRA. 

Again,  add  together  the  monomials  J 

The  result  after  reducing  (Art.  25),  is 

3.  Let  it  be  required  to  find  the  sum 
of  the  expressions 

Their  sum,  after  reducing  (Art.  25)  is 

27.  As  a  course  of  reasoning  similar  to  the  above  would 
apply  to  all  polynomials,  we  deduce  for  the  addition  of 
algebraic  quantities  the  following  general 

RULE. 

I.  Write  down  the  quantities  to  be  added  so  that  the  similar 
terms  shall  fall  in  the  same  column,  and  give  to  each  term  its 
proper  sign. 

II.  Reduce  the  similar  terms,  and  after  these  results,  write, 
with  their  propter  signs,  the  terms  which  cannot  be  reduced. 

EXAMPLES. 

1.  What  is  the  sum  of  Sax,  5ax,   — 2ax,  and   I3ax.  1 

Ans.   19ax. 

2.  What  is  the  sum  of  4ab  -f-  Sac  and  2ab  —  7ac  -f-  dl 

Ans.  Gab  +  ac  -j-  d. 

3.  Add  together  the  polynomials, 

3a2  —  262  -  4ab,  5a2  —  b2  +  2ab,  and  Sab  —  3c2  —  262. 

The  term  3a2  being  similar  to  f  „,„        . .,       _,,„ 

P       .  Sji2  —  4<p,b  —  2b2 

5a2,  we  write  8a2  for  the  result  _,„  .  rtl7         TO 

'  .  op-  +-  2«o  —    o2 

of    the    reduction    of    these    two<  ,   „,,       ~.2       „a 

terms,  at  the  same  time  slightly       Qg2  +    a&  _  552  _  3^ 

crossing  them,  as  in  the  first  term.  [_ 

27.  Give  the  rule  for  the  addition  of  Algebraic  quantities. 


AUDITION.  45 

Passing  then  to  the  term  —  4a6,  which  is  similar  to  4  2a6 
and  +  3a6,  the  three  reduce  to  +  ab,  which  is  placed  aftei 
8tt2,  and  the  terms  crossed  like  the  first  term.  Passing 
then  to  the  terms  involving  62,  we  find  their  sum  to  be 
—  562,  after  which  we  write   —  3c2. 

The  marks  are  drawn  across  the  terms,  that  none  of  them 
may  be  overlooked  and  omitted. 


(4)             (5) 

(6) 

en 

(3) 

a                  6a 

5a 

3a6 

Sac 

a                  ha 

56 

5a6 

8ac 

2a                11a 

5a  +  56 

8a6 

1  lac 

(9) 

(10) 

(11) 

labc  +  9ax 

Sax  +  36 

12a  —    6c 

-  3a6c  —  Sax 

5ax  —  96 

-  3a  —    9c 

4a6c  +  6ax  loax  —  66  9a  —  15c 

Note. — If  a  =  5,   6  =  4,    c  =  2,  x  ==  1,  what   are    the 
numerical  values  of  the  several  sums  above  found? 


(12) 

( 

13) 

(14) 

9a  +/ 

Gax 

—     8ac 

3a/ 4    9    +  m 

-Ga  +  ff 

—  7a* 

—    9ac 

ag  —  oaf  —  m 

-2a-/ 

ax 

+  17ac 

ab  —    ag  +  3g 

a  +  9 

0 

0 

ab  +  Ag 

(15) 

(16) 

7x  +  3a6  + 

3c 

8a;2 + 

9acx  +  13a262c2 

—  Sx  —  3a6  — 

5c 

—  7x2  - 

ISacx  +  14a262c2 

hx  —  9a6  — 

9c 

—  4a;2  + 

4acr  —  20a262c2 

9x  —  9ab  — 

11c 

(  17 

22h  -  3c  - 
Sh  +  8c  - 

) 
-V+%9 

-2f-9g  + 

5x 

19h  +  5c  - 

-9f- 

-Qg  + 

5x 

3x2+    0       +    7a262c2 

(18) 

19a/i2  +  3a364  —  Sax3 

—  Hah2  —  9a3b*  4  9aa:3 

2ah?  —  6a364  4     ax3 


46  ELEMENTAKY     ALGEBRA. 

(  19  )  (  20  ) 

7x  —  9y  +  5z  -f  3  —    g  8a  +    6 

~-    x  —  3y            —  8  —    g  2a  —    6  -f    c 

~    ^r    y-3z+l+7^  —  3a +6            4-  2<i 

—  2a;  +  6y  '+  3s  —  1  —    g  —  66  —  3c  -f  3d 

a;  +  8y  —  5z  +  9  —    g  —  5a  -f-  7c  —  8a7 

4a;  +  3y  +  0    +4  +  5y  ~2a  -  56~+~5c~^-~  3j 

21.  Add  together    —  b  +  3c  —  d  —  115e  f  6/  —  5y,    36 

—  2c  -  3a7  —  e  +  27/,     5c  -  8rf  +  3/  -  7y,     -  76  -  6c 
H-  17a7  +  9e  —  5/+  lly,     —  36  —  bd  -  2e  +  6/-  9#  +  A. 

^res.   —  86  —  109c  +  37/—  lOy  4-  A. 

22.  Add  together  the  polynomials,    7a26  —  3a6c  —  862c 

—  9c3  4"  cd2,     8abc  —  5a26  4-  3c3  —  462c  4-  cd2     and     4a26 

—  8c3  4-  962c  —  3d3. 

Ans.   6a26  4-  5a6c  —  362c  —  14c3  4-  2cd2  —  3d3. 

23.  What  is  the  sum  of,    5a26c  4-  66a;  —  4a/     —  3a26c 

—  66.T  4-  14a/    —  a/  4-  96a;  4-  2a26c,    4-  6a/  —  86a;  4-  6a26c. 

Ans.   10a26c  4-  6a;  4"  15a/ 

24.  What  is  the  sum    of,     a2n2  -\-  3a3m  4-6,      —  Qahi2 

—  6a3m  —  6,    4-96  —  9a3m  —  5a2w2. 

Ans.    —  10a2n2  —  12a3?ft  4-  96. 

25.  What     is    the    sum    of,     4a362c  —  16a4a;  —  9aa:3o', 
_}_  6a362c  —  Gax3d  +  17a4a;,    4-  leaa:^  —  a*x  —  9a362c. 

Ans.  a3b2c  -\-  ax3d. 

26.  What  is  the  sum  of,    —  Ig  4-  36  4-  4y  —  26,   4-  3g 

—  36  4-  26.  Ans.  0. 

27.  What   is   the    sum  of,    ab -\- 3xy — in  —  n,     — 0>xy 
-3m+  Un  +  cd,    4-  3xy  4-  4m  —  lOn  +  fg, 

Ans.  ab  4-  cd  4-  fg. 

28.  What  is  the  sum  of,    Axy  -\-  n  -\-  Gax  4-  9a?ft,    —  iixy 
■\-  Gn  —  Oaa;  —  8awi,    2xy  —  7n  -\-  ax  —  am.       Ans.   4-  aa\ 


SUBTRACT/ON.  47 

'29.  Add    the    polynomials      19a2£36  —  12a3c6,     5a2x36 
+.  I4a*cb  —  IQaz,      —  2a2x3b  —  12a3c6,     and    —  I8a2c36 

—  12a3c6  -j-  9ax.  Ans.  4a2x2b  —  22a3cb  —  ax. 

30.  Add  together    3«  +  b  +  c,      5a  -f  26  +  Sac,     a  +  c 
4  ac,  and    —  3a  —  9ac  —  86.       -4 /is.  Ga  —  56  +  2c  —  5ac. 

31.  Add    together    5a26  +  Qcx  -f  96c2,      7c.r  —  8«26,  and 
-  15ce  —  96c2  -(-  2a26.  Ans.   —  a26  —  2cx. 

32.  Add  together    8ax  +  hab  +  3a262c2,      —  18a*;  +  Qa2 
■f  10a&,    and    10ax  —  15a6  —  Ca262c2. 

Ans.    —  3a262c2  -f-  Ga2. 

33.  Add    together     3a2  +  5a2b2c2  —  9a\    la2  -  8a2b2c* 

—  10a3ar,    and    10a6  +  10a262c2  -f  19a3*. 

Ans.   IQa2  +  13a262c2  -f  10a6. 


SUBTRACTION. 

28.  Subtraction,  in  Algebra,  is  the  process  of  rinding  the 
simplest  expression  for  the  difference  between  two  alge- 
braic quantities. 

Thus,  the  difference  between  Ga  and  3a  is  expressed  by 
Ga  —  3a  =  3a  ; 
and  the  difference  between   7a36  and    3a36  by 
7a36  -  3a36  =  4a36. 

In  like  manner,  the  difference  between  4a  and  36,  is 
expressed  by   4a  —  36.     Hence, 

If  the  quantities  are  positive  and  similar,  subtract  the  co- 
efficients, and  to  their  difference  annex  the  literal  part.  If 
they  are  not  similar,  place  the  minus  siyn  before  the  quantity 
to  be  subtracted. 

28.  What  is  subtraction  in  Algebra?  How  do  you  find  this  differ- 
ence -when  the  quantities  are  positive  and  similar  ?  When  the}  are  not 
similar,  how  do  ywi  espreaa  the  difference  ? 


4S  ELEMENTARY     ALGEBKA. 

(1)  (2)  (3) 

From  Sab  Qax  9abc 

take  2ab  Sax  labc 

Rem.  ab  Sax  2abc. 

(4)  (5)  (6) 

From  \QaWc*  17a3b3c  24a2b2x 

take  9a2b2c  Sa3Pc  laWx 

Rem.  ~laWc  \4aWc  llaWx. 

(7)  (8)  (9) 

From  Sax  4abx  2am 

take  8c  9ac  ax 

Rem.         Sax  —  8e  4abx  —  9ac  2am  —  ax. 

29.  Let  it  be  required  to  subtract  from    4a 

the  binomial 26  —  3c 

The  difference  may  be  put  under  the  form  4a  —  {2b  —  3c) 
We  must  now  remark  that  it  is  the  difference  between  2b 
and  3c  which  is  to  be  taken  from  4a. 

If  then,  we  write 4a  —  2b, 

we  shall  have  taken  away  too  much  by  the  units  in  3c ; 
hence,  3c  must  be  added,  to  give  the  true  remainder,  which 
is 4a  —  2b  -f-  3c. 

To   illustrate    this    example   by  figures,  suppose   a  =  5, 
6  =  5,  and  c  =  3. 

We  shall  then  have 4a  •.-_  20 

and 2b  —  Se  =  10  —  9    =    1 

which  may  be  written     4a  —  (26  —  3c)  —  20  —  1    —  19. 


29.  If  26  —  3c  is  to  be  taken  from  4a,  -what  is  proposed  to  Le  done  ? 
If  you  subtract  26  from  4a,  have  you  taken  too  much  ?  How  tlier, 
must  you  supply  the  deficiency  3 


SUBTRACTION.  40 

Here  it  is  required  to  subtract  1  from  2l>.  If,  then,  we 
subtract  26  =  10,  from  4a  =  20,  it  is  plain  that  we  shall 
have  taken  too  much  by  3c  =  9,  which  must  therefore  be 
added  to  give  the  true  remainder. 

30.  Hence,  for  the  subtraction  of  algebraic  quantities,  we 
ha  ye  the  following  general 

RULE. 

I.  Write  the  quantity  to  be  subtracted  under  that  from  which 
it  is  to  be  taken,  placing  the  similar  terms,  if  there  are  any,  in 
the  same  column. 

II.  Change  the  signs  of  all  the  terms  of  the  subtrahend,  or 
conceive  them  to  be  changed,  and  then  reduce  the  polynomial 
result  to  its  simplest  form. 


(1) 

From  6ac  —  5a  6  +    c2 
Take  3ac  +  3a6  +  7c 

EXAMPLES. 

flj.                (i) 

»  o  a          6ac  —  5a6  +    c2 

s  m  a 

5  &-      —  3ac  —  3a6  —  7c 

Rem.  Sac  —  8ab  +    c2  — 

(2) 

From      6ax  —  a  +  362 
Take        9ax  —  a:  +    62 

•  7c.  £  »  g,s       3ac  -  8a6  +    c2  -  7ft 

1=8, 

(3) 

6yx  —  3a:2  +  56 
yx  —  3     +    a 

Rem.  —  Sax  —  a  +  x+  2b2.              5yx  —  Sx2  +  3  -f  56  —  a. 

(4)  (5) 

From      5a3  — 4a26  +    362c  4a6  —   ca*+3a2 

Take   —  2a3  +  3a26-    862c  5a6  -  4cd  +  3a2  +  562. 

Rem.       7a3  — 7a26-f  H62c.  —   a6  +  2>cd  —  5b\ 


30.  Give  the  rule  for  the  subtraction  of  Algebraic  quantities 
3 


50  ELEMENTARY     ALOSiiKA. 

6.  From    Gam  -f-  y   take  oam  —  x.     Ans.  3am  -f  x  -f  y. 

7.  From    3aa:   take   3as  —  y.  Ans.    ■+■  y. 

8.  From    7a262  —  a;2   take    18a262  +  x2. 

Ans.   -  lla262  —  2xz. 

9.  From     -  7/  +  3j»  —  8a;   take     —  6/  —  5m  —  2j  + 
3tf  +  S.  Ans.   —  f  +  8*re  —  Ox  —  3c/  —  8t 

10.  From    —  a  —  56  +  7c  —  d    take   46  —  c  +  2a7  +  2*. 

Ans.   —  a  —  96  +  8c  —  3d  —  2k. 

11.  From   .  .  —  3a+  6  —  8c+  7e  —  5/+  3A  -  Ix  —  13y 
take  &  +  2a  —  9c  +  8e  —  Ix  -f  7/  —  y  —  2,1  —  k. 

Ans.   —  5a-f-6  +  c  —  e  —  12/+  3h  —  12y  +  31. 

12.  From    a  -f-  6    take   a  —  b.  Ans.  26. 

13.  From    2z  —  4o  —  26  +  5    take    8  —  56  4-  a  +  Gx. 

Ans.   —  4z  —  5a  -f-  36  —  3. 

14.  From    3a  4-64-  c  —  d  —  10    take   c  4-  2a  —  a\ 

vl?is.  a  +  6  —  10. 

1 5.  From   3a  +  b  +  c^d  —  10   take  6  —  19  +  3a. 

Ans.  c  —  d  +  9. 

16.  From   2a6  4-62  — 4c  +  6c  — 6    take  3a2  —  c  4- 62. 

Ans.  2ab  —  3a2  —  3c  +  6c  —  6. 

17.  From   a3  4-  362c  4-  a62  —  a6c  take   63  +  a62  —  abc. 

Ans.  a3  4-  362c  —  63. 

18.  From    12a;  4-  6a  —  46  4-  40  take  46  —  3a  4-  4x  4-  Gd 
—  10.  Ans.  8x  +  9a  —  86  —  Gd  +  50. 

19.  From  2x  —  3a  4-  46  4-  6c  —  50  take  9a  +  x  4-  66  — 
6c  —  40.  Ans.  x  —  12a  —  26  +  12c  —  10. 

20.  From  6a  —46  —  12c  4-  12z  take  2x  —  8a  4-  46  —  6c. 

Ans.  14a  —  86  —  6c  4-  10ar. 

21.  From    8a6c  •-  1263a  +  Gcx  —  7xy    take   7c.c  —  xy  — 
I363a.  Ans.  Sale  4-  63a  —  ex  —  Gxy. 


SUBTRACTION. 


51 


31.  Polynomials  may  be  subjected  to  certain  transforma- 
tions, by  the  rule  for  subtraction. 

First  example,     .     .  6a2  —  Sab   -j-  2b2    —  26c, 

becomes 6a2  —  (Bab   —  2b2    +  26c). 

Second  .'    .     .     .    ".  7a3  —  8a2b  —  4b%  -f  6b2, 

becomes  .     .     ,     .     .  7«3  —  (8a2b  +  462c  —  662), 

or,  again, 7a3  —  8a26  —  (462c  —  6b2). 

Third  .     .     .     .     .  8a3  -  762    +    c     —  d, 

becomes  .....  8a3  —{7b2    —    c     -+-  d). 

Fourth  .     .     .     .     .  9b3  —    a     +  3a2    —  d, 

becomes  .     .     .     .     .  963  —  (a     —  8a2    -f  d). 

32.  Remark. — From  what  has  been  shown  in  addition 
and  subtraction,  we  deduce  the  following  principles. 

1st.  In  algebra,  the  term  add  does  not  always,  as  in  arith- 
metic, convey  the  idea  of  augmentation  ;  nor  the  term  sum, 
the  idea  of  a  number  numerically  greater  than  any  of  the 
numbers  added.  For,  if  to  a  we  add  —  b,  we  have  a  —  b, 
which  is,  properly  speaking,  a  difference  between  the  num- 
ber of  units  expressed  by  a,  and  the  number  of  units  ex- 
pressed by  b.  Consequently,  this  result  is  numerically  less 
than  a.  To  distinguish  this  sum  from  an  arithmetical  sum, 
it  is  called  the  algebraic  sum. 

Thus,  the  polynomial  2a2  —  3a26  -f  362c  is  an  algebraic 


31 1  How  may  you  change  the  form  of  a  polynomial  ? 
32.  In  algebra  do  the  words  add  and  su7n  convey  the  same  ideas  aa 
in  arithmetic  ?     What  is  the  algebraic  sum  of  9  and   —  4  ?     Of  8  and 

—  2  ?  May  an  algebraic  sum  ever  be  negative  ?  What  is  the  sum  of  4 
and  —  8  ?  Does  the  word  subtraction,  in  algebra,  always  convey  the 
idea  of  diminution  ?     What  h  the  algebraic  difference  between  8  and 

—  4  ?     Between  a  and  —  b  < 


b2  ELEMEHTART     ALGEBRA. 

sum  of  the  monomials  2a2,  —  Sa2b,  +  %b2c,  with  then 
respective  signs ;  but,  in  its  numerical  acceptation,  it  is  the 
arithmetical  difference  between  the  sum  of  the  units  con- 
tained in  the  additive  terms,  and  the  sum  of  the  units  con- 
tained in  the  subtractive  terms. 

It  follows  from  this,  that  an  algebraic  sum  may,  in  the 
numerical  applications,  be  reduced  to  a  negative  number,  or 
a  number  affected  with  the  sign  — . 

2d.  The  word  subtraction,  in  Algebra,  does  not  always 
convey  the  idea  of  diminution  ;  nor  the  term  difference,  the 
idea  of  a  number  numerically  less  than  the  minuend  :  for, 
the  numerical  difference  between  +  a  and  —  b  being  a  +  b, 
exceeds  a.  This  result  is  an  algebraic  difference,  and  can  be 
put  under  the  form  of 

a  —  (  —  b)  =  a  4-  b. 


MULTIPLICATION. 

33.  If  a  man  earns  a  dollars  in  one  day,  how  much  will 
he  earn  in  6  days  1  Here  it  is  simply  required  to  take  the 
number  a,  6  times,  which  gives  6a  for  the  amount  earned. 

1.  What  will  ten  yards  of  cloth  cost,  at  c  dollars  per  yard? 

Ans.   10c  dollars. 

2.  What  will  d  hats  cost,  at  9  dollars  per  hat  ? 

Ans.   9d  dollars. 

3.  What  will  b  cravats  cost,  at  40  cents  each? 

Ans.  405  cents 
■i.  What  will  b  pair  of  gloves  cost,  at  a  cents  a  pair  1 

33.  If  a  man  earns  a  dollar?  in  1  day,  how  much  will  he  earn  in  4 
Jays?  In  5  days?  In  8  days?  In  12  days  \  If  he  earns  c  dollars  a 
day.  how  much  will  he  earn  in  d  days  ?     What  is  multiplication? 


MUL'til'LlOAIIOH.  53 

Here  it  is  plain  that  the  cost  will  be  found  by  repeatir.g  6 
as  many  times  as  there  are  units  in  a :  Hence,  the  cost  is 
ab  cents.     Hence,  we  infer  that, 

Multiplication,  in  Algebra,  is  the,  process  of  taking  one 
quantity,  called  the  multiplicand,  as  many  times  as  there  are 
units  in  another,  called  the  multiplier. 

34,  If  a  man's  income  is  3a  dollars  a  week,  how  much 
will  it  be  in  46  weeks  ]  Here  we  must  repeat  3a  dollars  as 
many  times  as  there  are  units  in  45  weeks ;  hence,  the  pro- 
duct is  equal  to 

3a  X  46  =  12a6. 

If  we  suppose  a  =  4  and  6  =  3  the  product  will  be  equal 
to  144. 

Remark. — It  is  plain  that  the  product  12a6  will  not  be 
altered  by  changing  the  arrangement  of  the  factors ;  that 
is,  12a6  is  the  same  as  a6  x  12,  or  as  6a  X  12,  or  as 
ax  12  x  6  (See  Arithmetic,  §  26). 

35.  Let  us  now  multiply  3a262  by  2a2  6,  which  may  be 
placed  under  the  form 

3a262  X  2a26  =  3  X  2aaaabbb  ; 
in  which  a  is  a  factor  four  times,  and  6  a  factor  three  times : 
hence  (Art.  13). 

3a262  X  2a26  =  3  x  2aaaabbb  =  6a463, 

in  which,  we  multiply  the  co-efficients  together,  and  add  the 
exponents  of  the  like  letters. 

34.  Will  a  product  be  altered  by  changing  the  arrangement  of  the 
factors?  Is  Sab  the  same  as  36a?  Is  it  the  same  as  a  X  36?  As 
b  X  3a? 

35.  In  multiplying  monomials,  what  operation  do  you  perform  on  the 
to-efficients  ?  What  do  you  do  with  the  exponents  of  the  commou 
letters  ?     Wl  at  is  th«  rule  for  the  multiplication  of  monomials  ? 


54 


S  L  E  M  E  U  T  A  H  V     ALliEBHA. 


Hence,  for  the  multiplication  of  monomials,  we  have  the 
following 


RULE. 


I.  Multiply  the  co-efficients  together  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient  all  the  letters  which  enter 
into  the  multiplicand  and  multiplier,  affecting  each  with  an 
exponent  equal  to  the  sum  of  its  exponents  in  both  factors* 


EXAMPLES. 


1. 
2. 
3. 


Multiply 

by 


8a26c2  X  7abd2  =  56a3b2c2d\ 
2la3b2cd  X  8abc3  =  lG8a463c4rf. 
4abc  X    Idf  =z    2Sabcdf. 


(4) 

Sa2b 
2a2b 
Ga'b2 


(5) 

12a2 x 

12x2y 

144a2x3y 


(?) 

a2xy 

2xg2 
2a2x2y3 


(8) 

3ab2c3 

9a2b3c 

27aW 


627/2 
ay2z 
Gaxy3z2. 

(9) 

87cu'2y 
8&3.iV 
2Gla6^V- 


10.  Multiply    5a3£2£2  by  Gc5a;6.  ^4 /is.  SOa^V5^ 

11.  Multiply  10a*b5c8  by  7acc/.  ^4/is.  70a5b5c9d. 

12.  Multiply    9a3tey  by  OuVvxy.  Arts.  8laeb2x2y2. 

13.  Multiply  36a867c6^5  by  20ab2c3d\  Ans.  720u96;W9 

14.  Multiply  21axyz  by  9a2b2c2d2xyz. 

Ans.  2\2>a3b2c2d2x2y2z\ 

15.  Multiply  X&aWe  by  8abiy.  Ans.  104a*b3cxy, 


MULtlPLlCATlON  55 

Ifi.  Multiply  20a5b5cd  by  I2a2x2y.        Ans.  2AQa~b*cdx2y. 

17.  Multiply  14a466rf+y  by  20a3c2x2y. 

Ans.  2S0a'b6c2d*x2y\ 

18.  Multiply  8a353?/4  by  la4bxy5.   %  Ans.  biSa?bAxy*. 

19.  Multiply  Ibaxyz  by  5a5bcdx2y2.     Ans.  SI '  ha%cdx3y3z. 

20.  Multiply  51a2y2z2  by  9a2bc2x5y.       Ans.  459a46c3#T$s. 

21.  Multiply  2a3b2y2  by  18a&e.  ^4/w.  3Ua46:%2. 

22.  Multiply  64a3//i5a4^;  by  Sab2c3. 

Ans.  5\2a*b2c3m''xiyz. 

23.  Multiply  9a262c2d3  by  12a364c6.  Ans.  108a-WW3. 

24.  Multiply  2l6a£W8  by  3a362c5.         Ans.  648a469c8d8. 

25.  Multiply  70a867c4t/2/a;  by  \2a'¥c3dx2y3. 

Ans.  840a16612cVya;3y3. 

36.  We  will  now  consider  the  most  general  case  of  two 
polynomials. 

Let  a  represent  the  sum  of  all  the  additive  terms  of  the 
multiplicand,  and  —  6  the  sum  of  the  sub  tractive  terms. 
Let  c  denote  the  sum  of  the  additive  terms  of  the  multi- 
plier, and  —  d  the  sum  of  the  subtractive  terms.  The  mul- 
tiplicand may  then  be  represented  by  a  —  6,  and  the  mul- 
tiplier by  c  —  d  :  It  is  required  to  take  a  —  b  as  many  times 
as  jnere  are  units  in  c  —  d. 

Let  us  first  take  a  —  b  as  many 

times  as  there  are  units  in  c.  a   —  6 

We  begin  by  writing  ac,  which  is  c   —  d 

too  great,  by  b  taken  c  times ;  for,  ac  —  be 
it  is  only  the  difference  between  a  —  ad  +  bd 


and  b  which  is  to  be  taken  c  times.         ac  —  be  —  ad  -\-bd. 
Hence,  ac  —  be  is    the   product   of 
a  —  b  by  c. 

But  it  was  proposed  to  take  a  —  b  only  as  many  times  as 
there  are  units  in  the  difference  between  c  and  d ;  hence,  the 


56  ELEMENTARY     ALGEBRA. 

last  product  ac  —  be  is  too  large  by  a  —  b  taiien  d  times. 
But  a  —  b  taken  d  times,  is  ad  —  bd.  Subtracting  this  pro- 
duct from  ac  —  cb  (Art.  30),  and  we  have 

(a  —  b)  X  (o* —  d)  =  ac  —  be  —  ad  -f-  bd. 

37.  Hence,  we  have  the  following  rule  for  the  signs. 

When  two  terms  of  the  multiplicand  and  multiplier  are 
affected  with  like  signs,  the  corresponding  product  is  affected 
with  the  sign  -f-  ;  and  when  they  are  affected  with  contrary 
signs,  the  product  is  affected  with  the  sign  — . 

Therefore,  we  say  in  algebraic  language,  that  -j-  multi- 
plied by  -f ,  or  —  multiplied  by  — ,  gives  +  ;  —  multiplied 
by  +,  or  +  multiplied  by  — ,  gives  — . 

Hence,  for  the  multiplication  of  polynomials  we  have  the 
following 

KTJLE. 

Multiply  all  the  terms  of  the  multiplicand  by  each  term  cf 
the  multiplier,  observing  that  in  each  multiplication  like  signs 
give  plus  in  the  product,  and  unlike  signs  minus.  Then  reduce 
the  polynomial  result  to  its  simplest  form. 

EXAMPLES    IN    WHICH    ALL    THE    TERMS    ARE    PLU3. 


1.  Multiply        .... 

by      

The  product,  after  reducing, 

3a2  +     4a6  +  b2 

2a   -f-    bb 

6a3  -f    8a2b  -f  2ab2 

+  15a26  +  20ab2  +  5&3 

becomes     .... 

6a3  +  23a2b  -f  22a62  +  5b3. 

37#  "What  does  +  multiplied  by  +  give?  +  multiplied  by  — 
-  multiplied  by  +  ?  —  multiplied  by  —  ?  Give  the  rule  for  tb» 
nultiplication  of  polynomials. 


MULTIPLICATION.  57 

2.  Multiply  x2  +  2ax  -f  a2  by  x  +  a. 

Ans.  x3  +  3aa;2  +  3a2x  +  «3 

3.  Multiply  a;3  +  y3  by  x  +  y.  -4?is.  a:4  +  xy3  +  a;3y  +2/4 

4.  Multiply  Sab2  +  6a2c2  by  Zab2  +  3a2c2. 

ira*.  9a264  +  27a3b2c2  +  18a4c* 

5.  Multiply  a2i2  +  c2tf  by  a  +  6. 

^4ras.  a362  -j-  ac2G?  +  a263  -\-bc2d 

6.  Multiply  3aa;2  +  9ab3  +  c^5  by  6a2c2. 

Ans.  18a3c2x2  +  54a3c263  4-6a2c3d* 

7.  Multiply  64a3z3  4-  27a2a;  -f  9ab  by  8a3«f. 

-in*.  512a6ccte2  +  216a5«fo  +  72a46cd 

8.  Multiply  a2  +  2aa;  -f  a;2  by  a  +  x. 

Ans.  a3  +  3a2#  -f  3ax2  +  xl 

9.  Multiply  a3  +  3a2.r  +  Saa;2  +  x3  hy  a  -\-  x. 

Ans.  a4  +  4a3#  +  6a2a;2  4-  4aa;3  +  %* 

10.  Multiply  a;2  +  y2  by  a;  +  y. 

Ans.  x3  +  #y2  -4-  #2y  4-  y3 

11.  Multiply  a;5  +  «y8  +  7aa;  by  ax  +  5ax. 

^4  «s.  6aa;6  4-  6aa;2y6  +  42a2#2 

12.  Multiply  a3  4-  3a25  +  Sab2  +  b3  by  a  +  b. 

%  Ans.  a*  4-  4a35  +  6a262  4-  4a63  +  64 

13.  Multiply  a;3  4-  x2y  -{-  xy2  -\-  y3  by  x  -\-  y. 

Ans.  a;4  4-  2a;3y  +  2a;2y2  4-  2#y3  -j-  y* 

14.  Multiply  a;3  4-  2a;2  +  a;  +  3  by  3a;  4-  1. 

Ans.  3a;4  4-  7a;3  4-  5a;2  4-  10a   |-  3 

GENERAL    EXAMPLES. 

1.  Multiply 2aa;  —  Sab 

by 3a;    —     6. 


The  product 6aa;2 —  9a6a; 

becomes  after —   2abx  4-  tab3 


reducing 6aa;2— lla&a;  4*  3vt52 

3* 


R8  ELEMENTARY     ALGEBRA. 

2.  Miltiply  a4  —  263  by  a  —  b. 

Ans.  a5  —  2a  63  -  a45  +  26* 

3.  Multiply  a;2  —  3x  —  7  by  a:  —  2. 

,4 /is.  a;3  —  5a;2  —  x  -f-  14 

4.  Multiply  3a2  —  5a6  -j-  2b2  by  a2  —  7ub. 

Ans.  3a4  —  2Ga3b  +  37a2o2  —  14ao3. 
5    M  ultiply  b2  +  ¥  +  66  by  £2  —  1.  ^ns.  i8  -  b\ 

6.  Multiply  a:4  —  2a;3y  +  4x2y2  —  Sxy3  +  16y4  by  x-f  2y. 

Ans.  x5  +  32y5. 
7    Multiply  4a;2  —  2y  by  2y.  -4tjs.  8a-2y  —  4y2. 

8.  Multiply  2x  +  4y  by  2x  —  Ay.  Ans.  4a.-2  —  lGy2. 

9.  Multiply  x3  +  a;2^  -f-  a;y2  +  y3  by  x  —  y. 

Ans.  x*  —  y*. 

10.  Multiply  x2  -{-  xy  -\-  y2  by  a;2  —  xy  +  y2. 

-4«s.  a;4  +  x2y2  -f-  y*. 

11.  Multiply  2a2  —  3aa:  +  4.r2  by  5a2  —  Gux —2x2. 

Ans.  10u4  —  27a3.c  +  34a2z2  —  18a*3  —  8«*. 

12.  Multiply  3x2  —  2.ry  +  5  by  a;2  +  2xy  —  3. 

-4«s.  3a.4  +  4x3y  —  4x2  —  4a:2y2  -f  IGa-y  —  15. 

13.  Mulf'ply  3a;3  +  2x2y2  +  Sy2  by  2x3  —  3x2y2  4-  Sy3? 

j  Gx6  —  5a."V2  —  6afy*  +  Gx3y2  + 
(  15a.-3*/3  -  9x2y4  +  I0x2y5  +  15y5. 

14.  Mu^iply  Sax  —  Go6  —  c  by  2ax  +  a6  -f-  c. 

.Ins.  lGa2x2  —  Aa2bx  —  Ga2b2  +  Gacx  —  7abc  —  c2. 

15.  Multiply  3a2  —  562  +  3c2  by  a2  -  b2. 

Ans.  Su*  —  §a2b2  +  3a2c2  +  bb*  —  Sb2c2. 

1G.  3a2  —  5bd+    cf 

-  5a2  -j-  Aid  —  8c/. 


Pro. red.    -  15a*  +  37a2W— 2<da2cf-20b2d2  +  ±Abcdf-&cyi. 


MULTIPLICATION.  59 

38.  We  will  finish  the  subject  of  algebraic  multiplication, 
bf  making  known  a  few  results  of  frequent  use  in  Algebra. 

Let  it  be  required  to  form  the  square,  or  second  power, 
of  the  binomial  (a  -J-  b).     We  have,  from  known  principles, 

(a  +  b)2  =  (a '+  b)  (a  +  b)  =  a2  +  2ab  +  b2.     That  is, 

Tlie  square  of  the  sum  of two  quantities  is  equal  to  the  square 
uf  the  first,  plus  twice  the  product  of  che  first  by  the  second, 
plus  the  square  of  the  second. 

1.  Form  the  square  of  2a  -f-  ob.    We  have  from  the  rule 

(2a    +  U)2        =    4a2     +    12ab     4-    9b2. 

2.  (oab  +  Sac)2      =  2ba2b2  +    S0a2bc  +    9a2c2. 

3.  (5a2  +  8a26)2    =  25a4     +    80a46    +  G4a462. 

4.  (Qax  +  9a2^2)2  =  3t3a2x2  +  108a3z3  +  81a4.c4. 

39.  To  form  the  square  of  a  difference,    a  —  b,    we  have 

(a  -  b)2  ={a-b)  (a  —  b)=a2  —  2ab  +  b2 :    That  is, 

The  square  of  the  difference  between  two  quantities  is  equal  to 
lite  square  of  the  first,  minus  twice  the  product  of  the  first  by 
the  second,  plus  the  square  of  the  second. 

1.  Form  the  square  of  2a  —  b.     We  have 

(2a—  b)2  =  4a2  —  4ab  4-  b2. 

2.  Form  the  square  of  4ac  —  be.     We  have 

(4ac  —  be)2  =  I6a2c2  —  Sabc2  4-  b2c2. 

3.  Form  the  square  of  7a2b2  —  12ab3.     We  have 
(7a2i2  _  i2ai:!)2  =  49a464  —  168a365  4-  144a266. 

'68.  What  is  the  square  of  the  sum  of  two  quantities  equal  to  ? 

Hd,  What  is  the  square  of  the  difference  of  two  quantities  equal  to  ? 


60  ELEMENTARY     ALGEBRA. 

40.  Let  it  be  required  to  multiply  a  -f  6  by  a  —  b 
We  have 

(a  +  6)  X  (a  —  b)  =  a2  —  62.     Hence, 

The  sum  of  two  quantities,  multiplied  by  their  difference,  it 
equal  to  the  difference  of  their  squares. 

1.  Multiply  2c  +  b  by  2c  —  b.     We  have 

(2c  4-  b)  X  (2c  -  6)  =  4c2  -  62. 

2.  Multiply  9ac  +  36c  by  9ac  —  36c.     We  have 

(9ac  +  36c)  (9ac  -  36c)  =  81a2c2  -  962c2. 

3.  Multiply  8a3  +  7a62  by  8a3  —  7a62.     We  have 

(8a3  +  7a62)  (8a3  —  7a62)  =  64a6  —  49a26*. 

FACTORING    POLYNOMIALS. 

41.  It  is  sometimes  convenient  to  find  the  factors  of  a 
polynomial,  or  to  resolve  a  polynomial  into  its  factors. 
Thus,  if  we  have  the  polynomial 

ac  +  a6  +  ad, 

we  see  that  a  is  a  common  factor  to  each  of  the  terms  : 
hence,  it  may  be  placed  under  the  form 

a(c+  6  +d). 

1.  Find  the  factors  of  the  polynomial  a262  +  o?d  +  <*2f. 

Am.  a2(62  +  d+f). 

2.  Find  the  factors  of  3a26  +  6a262  +  b2d. 

Ans.  6(3a2  +  6a26  +  bd). 


40i  What  is  the  sum  of  t^o  quantities  multiplied  by  their  difference 
equal  to? 


DIVISION.  01 

3.  Find  the  factors  of  Sa2b  +  9a2c  +  18a2xy. 

Ans.  3a2  (b  +  3c  +  6zy). 

4.  Find  the  factors  of  8a?cx  —  l&acx2  -f  2ac5y  —  30a6c9#. 

Ans.  2ac(4aa;  —  9z2  +  c4y  —  15a5c8#). 

5.  Find  the  factors  of  a2  -j-  2ab  +  62. 

udws.  (a  +  6)  X  (a  +  &)• 

6.  Find  the  factors  of  a2  —  b2.     Ans.  (a  +  b)  X  (a  —  b). 

7.  Find  the  factors  of  a2  —  2ab  +  b2. 

Ans.  (a  —  b)  X  (a  —  b) 


DIVISION. 

42.  Division,  in  Algebra,  is  the  process  of  finding,  from 
two  algebraic  expressions,  a  third,  which  being  multiplied  by 
the  second,  will  give  a  product  equal  to  the  first.  The  first 
is  called  the  dividend,  the  second  the  divisor,  and  the  third, 
the  quotient. 

1.  The  division  of  72a5.  by  8a3  is  indicated  thus  : 
72a5 
8a3  ' 

It  is  here  required  to  find  a  third  monomial,  which,  mul- 
tiplied by  the  second,  will  produce  the  first.  It  is  plain  that 
the  third  monomial  is  9a2  :  Hence 

72a5 

— -5-  =  9a2  ;  for,  8a3  X  9a2  =  72a5. 
8a3 

The  quotient  9a2,  is  obtained  by  dividing  the  co-efficient 
of  the  dividend  by  the  co-efficient  of  the  divisor,  and  subtracting 
the  exponents  of  the  common  letter. 

42.  What  is  division  in  Algebra?  Give  the  rule  for  dividing  mono- 
mials. 


02 


ELKMENTAllY     ALGKBUA, 


Also, 
Again, 


B5a3b2c 
lab 

lab  x  5a26c  =  ZhaWc. 
56a462c2 


5a3_1  b2~lc  —  5a2bc. 


8a3bc 


=  labc. 


Hence,  for  the  division  of  monomials  we  have  the  fol- 
lowing 

KULE. 

I.  Divide  the  co-efficient  of  the  dividend  by  the  co-efficient 
of  the  divisor,  for  a  new  co-efficient. 

II.  Write  after  this  co-efficient,  all  the  letters  of  the  dividend, 
and  affect  each  with  an  exponent,  equal  to  the  excess  of  its 
exponent  in  the  dividend  over  that  in  the  divisor. 

From  this  rule  we  find 


48a3b3c2d       .  91    ,          150a5bscd3 
—  4a2bcd  ; 
12ab2c                               SQaWd2 

=  5a2b3cd. 

1. 

Divide  lGx2  by  8x. 

Ans.  2x. 

2. 

Divide  lbax2y3  by  Say. 

Ans.  bx2y2 

3. 

Divide  84aPx  by   1262. 

Ans.  labx. 

4. 

Divide  3Qa4b5c2  by  9a3b2c. 

Ans.  4ab3c. 

5. 

Divide  8  a3b2c  by  8a2b. 

Ans.  11  abc. 

6. 

Divide  99a4&4*5  by   lla3i2:c*. 

Ans.  9ab2x. 

7. 

Divide  108x6y5z3  by  54z52. 

Ans.  2xy°z2. 

8. 

Divide  04x7yV  by    lGx6y*s5. 

Ans.  4xyz, 

9. 

Divide  96«766c5  by   I2a2bc. 

Ans.  8a56sc*. 

10. 

Divide  54a1c5de  by  27  acd. 

Ans.  2aGcid5. 

11. 

Divide  S8a*b6d*  by  2a3bid. 

Ans.  19abd3. 

DIVISION, 


63 


12.  Divide  42a25V  by  labc.  Ans.  Qabc. 

13.  Divide  G4crW  by  32a*6c.  Am.  2ab3c\ 

14.  Divide  128a5^6y7   by  Waxy*.  Ans.  &a4x5y3. 

15.  Divide  I32bd5f*  by  2c/4/.  Ans.  QQbdfK 
1G.  Divide  256a46W7  by  16a3ic6.  Ans.  16ab3c2d\ 

17.  Divide  200a8?»2/i2  by  bOctPmn.  Ans.  4a?nn. 

18.  Divide  300^ V^2  ty  60xy2z.  ^ns.  bx2y2z. 

19.  Divide  27a5b2c2  by  9a6r.  ^4/is.  3a46c. 

20.  Divide  64a;^6s8  by  32nyV.  ^4/ts.  2a2^2. 

21.  Divide  88a5b6c8  by  llay6'c6.  Ans.  8a2b2c2. 

43.  It  follows  from  the  preceding  rule,  that  the  exact 
division  of  monomials  will  be  impossible, 

1st.  When  the  co-efficient  of  the  dividend  is  not  exactly 
divisible  by  that  of  the  divisor. 

2d.  When  the  exponent  of  the  same  letter  is  greater  in 
the  divisor  than  in  the  dividend. 

3d.  When  the  divisor  contains  one  or  more  letters  not 
found  in  the  dividend. 

When  either  of  these  three  cases  occurs,  the  quotient  may 
be  expressed  under  the  form  of  a  monomial  fraction  ;  that 
is,  a  monomial  expression,  necessarily  affected  with  the 
algebraic  sign  of  division.  Such  expressions  are  said  to 
be  in  their  simplest  form,  when  the  numerator  and  denomi- 
nator do  not  contain  a  common  factor. 

Tor  example,    12a46'W,  divided  by   8a2bc2,  gives 
12a4iW# 
8a26c2 


43.  What  is  the  first  case  named  in  which  the  division  of  monomials 
will  nut  be  exact  f  What  is  the  second  ?  What  is  the  third  ?  If  eiiher 
of  these  cases  occur,  can  the  exact  division  he  made  ?  Under  what  form 
tvi'd  the  quotieut  then  remain  ?  May  this  fraction  be  often  reduced  to  u 
simpler  form  t 


04  ELEMENTARY     ALGEBRA. 

which  may  be  reduced  by  dividing  the  numerator  and  deno- 
minator by  the  common  factors  4,   a2,   6,   and  c,   giving 

12a*b2cd         Sa2bd 


Also, 


8a26c2  2c 

25a*b2d3  5a 


15a*W         Sb*d 


44.  Hence,  for  the  reduction  of  a  monomial  fraction  to 
it',  simplest  form,  we  have  the  following 

RULE. 

Suppress  every  factor,  whether  numerical  or  literal,  that  is 
common  to  both  terms  of  the  f -action,  and  the  result  will  be 
the  reduced  fraction  sought. 

From  this  new  rule  we  find, 

(1)  (2) 

ASa3b2cd3      _  4ad2.         ,    37a  b3c5d     _  3762c  . 
36V6We  =  Sbce"    m        6a3b  c*d2  ~    6a2d  ' 

(3)  (4) 

,             7a2b  1  ,    4a2b2  2a 

also      .  ==  — :    and 


I4a3b2         2ab  Gab*  362 

76c- 


5.  Divide   49a252c6    by    14a36c*.  Ans. 

6.  Divide    Qamn   by    3a5c.  Ans 

7.  Divide    18a2b2mn2   by    12a*b*cd.  Ans. 


2a 
2mn 
~bc~ 
3mn2 


2a2b2cd 


44i  Give  the  rule  for  the  reduction  of  a  monomial  fraction. 


DIVISION.  G5 

7a*c6d 


8.  Divide  2Sa5bec7dB  by    IQabWm.  Am, 

9.  Divide  72a3c262  by    12<z5cW.  Ans. 

10.  Divide  100a8bsxnm  by  25a364o?.  Ans. 

11.  Divide  96a558c9i/  by   75a2cxy.  ^ns. 

12.  Divide  85m2»3/c2y3   by    lSamhif.  Ans. 

13.  Divide  127d3x2y2   by    lW^y4.  ^4ws.    , 

lodxy2 

45.  If  we  have  expressions  of  the  form 
a        a2      a3       a4       a5 
a        a2      a3       a*       a5 

and- apply  the  rule  for  the  exponents,  we  shall  have 


4b3m 

6 

a2c2bd 

4a5bxmn 

'         d      ' 

S2a3b8c*df 

25xy 

17  n2x2y3 

Sam2 

127 


—  =  a1  -  1  =  a0,   %-  =  a2"2  =  a0,    ~  =  a3~3=  a0,  &a 
a  a2  a3 

But  since  any  quantity  divided  by  itself  is  equal  to  1,  it 
follows  that 

—  =  a°  =  l,    %-  =  a2~2  =  a0=l,  &c., 
a  az 

or  finally,  if  we  designate  the  exponent  by  m,  we  have 

—  =  am~m  =  a0  =  1  :  that  is, 
am 

The  power  of  any  number  whose  exponent  is  0,  is  equal  to  1  ; 
and  hence,  a  factor  of  the  form  a0  may  be  omitted,  being 
equal  to  1. 

45.  What  is  a"  equal  to  ?     What  is  b"  equal  to  ?     What  is  the  power 
{A  any  number  equal  to,  when  the  exponent  of  the  power  is  0  ? 

4 


00  ELEMENTAKY     ALOEBEA, 

2.  Divide  Ga262c4J  by   2a2b2d. 

Scrota 

3.  Divide  8a*63c4^s  by  4a26W5.  Ans.  2a?. 

4.  Divide  16aW9  by  Sa%sd.  Ans.  2d8. 

5.  Divide  32»i3/i3.c2*/2  by   4m3n3xy.  Ans.  8xy. 

6.  Divide  96a*i5J8c9  by   24a46^5e9.  ^4/is.  46i3 

SIGNS  IN  DIVISION. 

46.  The  object  of  division,  is  to  find  a  third  quantity 
called  the  quotient,  which,  multiplied  by  the  divisor,  shall 
produce  the  dividend. 

Since,  in  multiplication,  the  product  of  two  terms  having 
the  same  sign  is  affected  with  the  sign  -j-,  and  the  product 
of  two  terms  having  contrary  signs  is  affected  with  the 
sign  — ,  we  may  conclude, 

1st.  That  when  the  term  of  the  dividend  has  the  sign  +, 
and  that  of  the  divisor  the  sign  of  +,  the  corresponding 
term  of  the  quotient  must  have  the  sign   +. 

2d.  When  the  term  of  the  dividend  has  the  sign  -f->  and 
that  of  the  divisor  the  sign  — ,  the  corresponding  term  of 
the  quotient  must  have  the  sign  — ,  because  it  is  only  the 
sign  — ,  which,  multiplied  with  the  sign  — ,  can  produce 
the  sign  +  of  the  dividend. 

46.  What  will  the  quotient,  multiplied  by  the  divisor,  be  equal  to  ? 
If  the  multiplicand  and  multiplier  have  like  signs,  what  will  be  the  sign 
of  the  product  ?  If  the}'  have  contrary  signs,  what  will  be  the  sign  of 
the  product  ?  When  a  term  of  the  dividend  and  the  term  of  the  divisor 
have  the  same  sign,  what  will  be  the  sign  of  the  corresponding  term  of 
the  quotient  ?  When  they  have  different  Bigns,  what  will  be  the  sign  of 
tbe  term  of  the  quotient  ? 


DIVISION.  07 

3d.  "When  the  term,  of  the  dividend  has  the  sign  — ,  and 
that  of  the  divisor  the  sign  +,  the  term  of  the  quotient 
must  have  the  sign  — .     Again  we  say  for  brevity,  that, 

+  divided  by  +,  and  —  divided  by  — ,  give  -f-  ;    ' 
—  divided  by  +,  and  +  divided  by  — ,  give  — . 


EXAMPLES. 

1.  Divide  4ax  by   —  2a.  Ans.  —  2x. 
Here  it  is  plain  that  the  answer  must  be  —2x',  for, 

—  2a  x  —  2x  =  +  4ax,  the  dividend. 

2.  Divide  oQa3x2  by  —  12a2x.  Ans.  —  Sax. 

3.  Divide  —  58a3b5c2d2  by  29a2b*c.  Ans.  —  2abccP. 

4.  Divide  —  84a*b5d3  by   —  42a2b2d.  Ans.  2a2bW. 

5.  Divide  Q4c4d5x3  by   lQc^dx.  Ans.  4dix?. 

6.  Divide  —88bix5y6  by  -  24b3cdx*.         Ans.  +  11&y<. 

ocd 

7.  Divide  77a4y324  by   —  lla4y3z4.  Ans.  —  7, 

8.  Divide  84aWc2d  by   —  42aib2c2d.  Ans.  —  2. 

9.  Divide   —  60a'b6c4d  by  '—  \2aBVcbd2.   Ans.  4-5- 

abed 

10.  Divide   —88aeb1c6  by  8a566c6.  Ans.  —Uab. 

11.  Divide  16a;2  by   —  8x.  Ans.  —  2x. 

12.  Divide   —  15a2xy3  by  Sat/.  Ans.  —5axy2. 

13.  Divide   -  84ab3x  by   —  12b2.  Ans.  7abx. 

14.  Divide   —  9Qa*b2c3  by    12a36c.  Ans.  —  Sabc2. 

15.  Divide   —  144a9i8c7^5  by  —  36a46<W.    Ans,  iaWcd*. 

16.  Divide  25Qa3bc2x3  by    —  16a2cz2.  Ans.  —]6abcx. 

17.  Divide   —  300a5&4c3.*2  by  30a463c2z.  Ans.  -  IQabcx. 

18.  Divide  500a869c6  by   —  100a768c*.  Ans.  —  5a£c2 


68  ELEMENTARY     ALOE  SKA. 

19.  Divide    —  64a5£8c7    by    —  8a467c5.  Ans.  8abc. 

20.  Divide    +  96aW9  by    —  2±aib2d.    Ans.  —  iab2d%. 

21.  Divide   72a563c74   by    —  8a*b2d.  Ans.  —  9ab<P. 

Division  of  Polynomials. 

FIRST   EXAMPLE. 

47.  Divide   a2  —  lax  -fa:2    by    a  —  x. 

It  is  found  most  convenient,  Dividend.       Divisor. 

in  division  in  algebra,  to  place  a2  —  2ax  -j-  x2  \a  —  x 

the  divisor  on  the  right  of  the  a2  —    ax  a  —  x 

dividend,  and  the  quotient  di-  —    ax  +  x2   Quotient, 

rectly  under  the  divisor.  —    ax  +  x2 

We  first  divide  the  term  a2,  of  the  dividend,  by  the  term 
a  of  the  divisor :  the  partial  quotient  is  a,  which  we  place 
under  the  divisor.  We  then  multiply  the  divisor  by  a,  and 
subtract  the  product  a2  —  ax  from  the  dividend,  and  to  the 
remainder  bring  down  x2.  We  then  divide  the  first  term 
of  the  remainder,  —  ax  by  a,  the  quotient  is  —  x.  We 
then  multiply  the  divisor  by  —  x,  and,  subtracting  as  before, 
we  find  nothing  remains.    Hence,  a  —  a;  is  the  exact  quotient. 

In  this  example,  we  have  written  the  terms  of  the  dividend 
and  divisor  in  such  a  manner  that  the  exponents  of  one  of  the 
letters  go  on  diminishing  from  left  to  right.  This  is  what  is 
called  arranging  the  dividend  and  divisor  with  reference  to 
that  letter.  By  this  preparation,  the  first  term  on  the  left 
of  the  dividend  is  the  one  which  must  be  divided  by  the 
first  on  the  left  of  the  divisor,  in  order  to  obtain  the  first 
term  of  the  quotient. 

47.  What  do  you  understand  by  arranging  a  polynomial  with  refer- 
ence to  n  particular  letter  ? 


division.  C$y 

48    Hence,  for  the  division  of  polynomials,  we  have  this 

RULE. 

I.  Arrange  the  dividend  and  divisor  with  reference  to  the 
same  letter,  and  then  divide  the  first  term,  on  the  left  of  the 
dividend  by  the  first  term  on  the  left  of  the  divisor,  the  result 
is  the  first  term  of  the  quotient ;  multiply  the  divisor  by  this 
term,  and  subtract  the  product  from  the  dividend. 

II.  Then  divide  the  first  term  of  the  remainder  by  the  first 
term  of  the  divisor,  which  gives  the  second  term  of  the  quotient ; 
multiply  the  divisor  by  the  second  term,  and  subtract  the  pro- 
duct from  the  result  of  the  first  operation,  Continue  the  same 
process  until  you  obtain  0  for  a  remainder,  or  until  the  first 
term  of  the  remainder  cannot  be  divided  by  the  first  term  of 
the  divisor. 

SECOND   EXAMPLE. 

Let  it  be  required  to  divide 

51a262  +  10a*  —  48a36  —  156*  +  4a63  by  4ab  —  5a2+362. 
We  here   arrange  with  reference  to  a. 

Dividend.  Divisor. 


10a*—  48a3b  +  51a262+   4ab3  — 156* 
10a*—    8a3Z>  —    6a262 

—  40a36  -f-  57a262+  4a&3  — 156* 

—  40a36  +  32a262+  24a63 

25a262— 20ai3-156* 
25a262- 20a63  — 156* 


|  —  5a2  +  4a5  +  362 
—  2a2  +  8ab  —  562 
Quotient. 


48^  Give  the  general  rule  for  the  division  of  polynomials  ?  If  the 
first  term  of  the  arranged  dividend  is  not  divisible  by  the  first  term  of 
the  arranged  divisor,  is  the  exact  division  possible  ?  If  the  first  term  of 
any  partial  dividend  is  not  divisible  by  the  first  term  of  the  divisor,  is 
the  exact  division  possible  ? 


70 


ELEMENTARY     ALGEBRA 


Remark. — When  the  first  term  of  the  arranged  dividend 
is  not  exactly  divisible  by  that  of  the  arranged  divisor,  the 
complete  division  is  impossible  ;  that  is  to  say,  there  is  not 
a  polynomial  which,  multiplied  by  the  divisor,  will  produce 
the  dividend.  And  in  general,  we  shall  find  that  the  exact 
division  is  impossible,  when  the  first  term  of  any  one  of  the 
partial  dividends  is  not  divisible  by  the  first  term  of  the 
divisor. 

GENERAL    EXAMPLES. 

1.  Divide  18a;2  by  9x.  Ans.  2z. 

2.  Divide  I0x2y2  by   —  bx2y.  Ans.  —  2y. 

3.  Divide  —  9ax2y2  by  9x2y.  Ans.   —  ay. 

4.  Divide  —  8a;2  by   —  2x.  Ans.   +  4a;. 

5.  Divide  IQab  +  15ac  by  5a.  Ans.  2b  +  8c. 

6.  Divide  oOax  —  54a;  by  6a*.  Ans.  5a  —  9. 

7.  Divide  10a;2y  —  15y2  —  5y  by  5y.  Ans.  2x2  -Sy  —  l. 

8.  Divide  12a -{- Sax  —  18ax2  by  3a.  Ans.  4  +  x  —  6x2. 

9.  Divide  6aa;2  4-  9a2a;  +  a2x2  by  ax.  Ans.  6x  -\-  (Ja  4-  ^- 

10.  Divide  a2  \~  2ax  -\-  x2  by  a  +  x.  Ans.  a  +  x. 

1 1.  Divide  a3  —  2>a2y  +  Say2  —  y3  by  a  —  y. 

Ans.  a2  —  2ay  +  y2. 

12.  Divide  24a26  —  \2a\b2  —  Gab  by   —  Qab. 

Ans.  —  4a  +  2a2cb  +  1. 

13.  Divide  Gx*  —  96  by  3a;  —  6.  Ans.  2a;3  4- 4a;2  4- Sa- 4- 16. 

14.  Divide   .     .      a5  —  5a4a;  4-  10a3x2  —  10a2z3  +  5ax*  — 
t5  by  a2  —  2aa;  4-  x2.  Ans.  a?  —  3a2a;  4-  Saa;2  —  a-3. 

15.  Divide  48a;3  —  76a«2  —  64a2x  4-  105a3  by    2a;  —  3a. 

Ans.  24a;2  —  2ax  —  35a2. 


DIVISION.  71 

16.  Divide   y6  —  oy^x2  +  3y2x4  —  sfi   by    y3  --  oi/?x  -j- 
2yx2  —  x3.  Ans.  y3  +  3y2a;  +  3yx2  -f-  ar3. 

17.  Divide    64a4Z>6  —  25u2b3   by    8a263  +  5a&4. 

Ans.  8a2b3  —  5«54. 

18.  Divide   6a3  +  23a26  +  22ab2  +  563    by    3a2  -(-  4«6 
f  62.  Jbw.  2a  -f  56. 

19.  Divide   6a#6  +  6ax2y6  +  42a2.r2   by   ax  -f  5a.r. 

u4ns.  x5  -f-  ^y6  +  7a.r. 

20.  Divide  —  15a4  +  S7a2bd  —  29a2cf-  20  W  -f  44&-07/ 
-  8c2/2   by    3a2  —  55c?  +  cf.         Ans.  —  5a2  +  Abd  —  Scf. 

21.  Divide   a4  +  x2y2  +  y4    by   a;2  —  xy  +  y2. 

.4ns.  a;2  -j-  a-'y  +  y2. 

22.  Divide  a;4  —  y4  by  x  —  y.  Ans.  x3  -f-  #2y  +  #y2  +  y3^ 

23.  Divide  3a4  —  8a262  +  3a2c2  -f  564  -  352c2  by  a2  —b\ 

Ans.  Sa2  —  fib2  +  3c2. 

24.  Divide  Gx6—5x5y2  -0x4y4  +  Gx3y2  +  15x3y3  —  9a;2y4 
-j-  10.c2y5  +  15y5  by  3x*  +  2.e2y2  +  3y2. 

Ans.  2x3  —  ox2y2  +  5y3. 

,25.  Divide  —  c2  +  16a2z2  —  labc  —  4a25a;  —  6a262+  Qacz 
by    Sax  —  Gab  —  c.  Ans.  2ax  +  ab  +  c. 

26.  Divide  .    .  Sx4  +  &e3y  —  4x2  —  4x2y2  +  lQxy  —  15 
by    2xy  +  x2  —  3.  ^4?i5.  3x2  —  2.ry  -f-  5. 

27.  Divide   &£  +  32y5   by   x  +  2y. 

J/i.<?.  x*  —  2x3y  +  4a.-2y2  —  8.ry3  +  IGy4. 

2*'.  Divide   3a*  —  20a36  —  14a63  -f  37a262    by    262  — 
&ih   «►  $a\  Am.  a2  ~  lab. 


72  ELEMENTARY     ALGEBRA. 


CHAPTER  EL 

Algebraic  Fractions. 

49.  Algebraic  fractions  are  of  the  same  nature  ai  *rith- 
metical  fractions ;  that  is,  we  must  conceive  that  some  unit 
one  has  been  divided  into  as  many  equal  parts  as  there  are 
units  in  the  denominator,  and  that  one  of  these  parts  is 
taken  as  many  times  as  there  are  units  in  the  numerator. 
Hence,  addition,  subtraction,  multiplication,  and  division, 
are  performed  according  to  the  rules  established  for  arith- 
metical fractions. 

It  will  not,  therefore,  be  necessary  to  demonstrate  these 
rules,  and  in  their  application  we  must  follow  the  procedures 
indicated  for  the  operations  on  entire  algebraic  quantities. 

50.  Every  quantity  which  is  not  expressed  under  a  frac- 
tional form  is  called  an  entire  quantity. 

51.  An  algebraic  expression  Vhich  is  partly  entire  and 
partly  fractional,  is  called  a  mixed  quantity. 

49.  How  are  algebraic  fractions  to  be  considered?  What  does  the 
denominator  show  ?  What  does  the  numerator  show  f  How  then  are 
the  operations  in  fractions  to  be  performed  ? 

50.  What  is  an  entire  quantity  ? 
61.  What  is  a  mixed  quantity  f 


ALGEBRAIC     FRACTIONS.  73 

CASE    I. 

To  reduce  a  fraction  to  its  simplest  form. 

52,  A  fraction  is  said  to  be  in  its  simplest  form,  when 
there  is  no  common  factor  in  the  numerator  and  denomi- 
nator. The  rule  for  reducing  a  monomial  fraction  to  its 
simplest  form  has  already  been  given  (Art.  44). 

With  respect  to  polynomial  fractions,  examples  under  the 
following»cases  are  easily  reduced. 

1.  Take,  for  example,  the  expression 

a2-52 
a2  —  2ab  +  b*  ' 

This  fraction  can  take  the  form 

(a  +  b)   (a  —  b) 

(a  -  by 

(Art.  39   and  40).     Suppressing  the  factor   a  —  b,  which 
is  common  to  both  terms,  we  obtain 

a  +  b 
a  —  b 

2.  Again,  take  the  expression 

5a3  —  10a25  +  5ab2 

8a3-  -  8a26 

This  expression  can  be  decomposed  thus: 

5a(a2  —  2ab  +  b2) 
8a2(a  —  b)  * 

5a  (a  -  b)2 

or —  • 

'  8a2(a  -  b) 


52i  How  do  you  reduce  a  fraction  to  its  simplest  terms ! 
4 


74  ELEMENTARY     ALGEBRA. 

Suppressing  the  common  factors  a(a  —  b),  the  result  is 


Hence,  to  reduce  any  fraction  to  its  simplest  form,  we  sup- 
press or  cancel  every  factor  common  to  the  numerator  and 
denominator 

Note. — Find  the  common  factors  of  the  numerator  and 
denominator  as  explained  in  (Art.  41). 

EXAMPLES. 

.     .p   ,  3a2  +  QaW    '  . 

1.  Keduce  ■      : -—-  to  its  simplest  form. 

12a*  +  6a3c2  r 

An,        l+™ 


n    T,   ,  15a5c  +  25a9d   4     ..       .       .     L   , 

2.  Keduce     __  _   ,   nn  „      to  its  simplest  form 

25a2  -f  30a2  r 

Ans. 

8567c<i5 

3.  Reduce  to  its  simplest  form. 

1567c8^5  r 


4a2  +  2ac2 
brm. 
3a3c  +  ba">d 


Ans. 


4.  Eeduce   ■ —   to  its  simplest  form. 

12c5d8f<> 


17 
3c7  * 


Ans, 


be 

"ay** 

B    _   ,           27a*6*  —  81a  66  .         3a3  -  962 

5.  Keduce   — - — = „„  „7J  •  Ans. 


63a  66  —  36a26*  ,        lb2  —  4a  ' 

96a362c 

6.  Reduce    tt-^t-  to  its  simplest  form.     Ans.  —  8. 

—  12a3o2c 

2465  —  36a&*  45  —  6a 

7-   Reduce   48a*6*-66a^  '  AlUm  8a*  -  lla«6* 


ALtlEBSAlO     FRACTIONS.  75 

CASE    II. 

53.  To  reduce  a  mixed  quantity  to  the  form  of  a  frao- 
tion. 

EULE. 

Multiply  the  entire  part  by  the  denominator  of 'the  fraction  ; 
add  to  this  product  the  numerator,  and  under  the  result  place 
the  given  denominator. 

EXAMPLES. 

1.  Reduce  Q>\  to  the  form  of  a  fraction. 

6  x  7  =  42  :     42  +  1  =  43  :     hence,     64  =  ~ . 

(a2  —  x2) 

2.  Reduce  x  — *  to  the  form  of  a  fraction. 

x 

a2  —  x2       x2  —  (a2  —  x2)       2x2  —  a2 

z =  ^ -  = Ans. 

xxx 

n%  -4—  x2 

3.  Reduce  x to  the  form  of  a  fraction. 

2a 

ax  —  ar? 


2a 


2x 7 

4.  Reduce  5  -\ - to  the  form  of  a  fraction. 

Sx 

a       17s  -  7 

Ans.  — • 

ox 

1 

5.  Reduce  1 to  the  form  of  a  fraction. 

a 

2a  —  x  -f  1 
Ans. . 


53   How  do  you  reduce  a  mixed  quantity  to  tho  form  of  a  fraction  t 


76 


ELEMENTARY     ALGEBRA. 

6.  Reduce     1  4-  2s —     to  the  form  of  a  fraction. 

5a; 

10x2+4x+3 
Ans. 


bx 

3c  4-  4 

7.  Reduce     2a  4-  b to  the  forai  of  a  fraction. 

o 

16a +  86  —  3c  —  4 
Ans.  - . 


8.  Reduee  Qaz  4-  h to  the  form  of  a  fraction, 

4a 

.        I8a2x  4-  5ab 
Ans. 


4a 

8  _f_  Qa2b2x* 

9.  Reduce  8  -f-  3a& ,^  ,-- —  to  the  form  of  a  fraction. 

I2ao:fc* 

9Qabx*  4-  30a2^4  —  8 
-4ns.    — 


12a6z* 

3§2  _  gc* 

10.  Reduee   9  4-   r— -  to  the  form  of  a  fraction. 

a—  b2 

.       9a  —  6b2  —  8c4 

Ans. = 

a  —  b2 


CASE    III. 

54.  To  reduce  a  fraction  to  an  entire  or  mixed  quantity. 

EULE. 

Apply  the  process  for  division  until  the  first  term  of  the 
remainder  is  not  divisible  by  the  first  term  of  the  divisor.  To 
the  quotient,  thus  obtained,  add  the  last  remainder  divided  by 
the  denominator. 

54.  Hrnv  do  you  reduce  a  fraction  to  an  entire  or  mixed  quantity? 


ALGEBRAIC     FRACTIONS.  77 


EXAMPLES. 


,    t>  j         8966 

1.  lieduce    — — —   to   an   entire  number. 

8 

8)8966( 


1120  .  .  .  6  rem. 

Hence,  1120g  =  Ans. 

2.  Reduce    to   a   mixed   quantity. 

a2 
Ans.  a . 

x 

ax  —  xz 

3.  Reduce    to  an  entire  or  mixed  quantity. 


Ans.  a  —x. 


4.  Reduce    ~ to  a  mixed  quantity. 


A  2«2 

Ans.  a . 

b   ' 

5.  Reduce   to  an  entire  quantity.     Ans.  a  +  x. 


6.  Reduce L_   to  an  entire  quantity. 

x      y 

Ans.  x2  -f-  xy  -j-  y2 

_    „   ,  10a:2  -  5x  +  2 

7.  Reduce to  a  mixed  quantity. 

uX 

3 

Ans  2x  —  1  -f-  — -. 
5a; 

a    _  36ar3  -  72a;  _j_  S2a2x2  . 

8.  Reduce to  a  mixed  quantity. 

.       0        32a2z 
Ans.  4tX2  —  8  H -r— 


78  ELEMENTARY     ALGEBRA. 

CASE    IV. 

55.  To  reduce  fractions  having  different  denominators  to 
equivalent  fractions  having  a  common  denominator. 

RULE. 

Multiply  each  numerator  into  all  the  denominators  except 
its  own,  for  new  numerators,  and  all  the  denominators  together 
for  a  common  denominator. 

EXAMPLES. 

1.  Reduce  \,  \,  and  f-,  to  a  common  denominator. 

1x3x5  =  15  the  new  numerator  of  the  1st, 
7  x  2  X  5  =  70  "  "         "        2d. 

4x3x2  =  24  "  "         "         3d. 

and     2  x  3  X  5  =  30  the  common  denominator. 

Therefore,   -|^,   |-g,   and   |^,  are  the  equivalent  fractions. 

Note. — It  is  plain  that  this  reduction  does  not  alter  the 
values  of  the  several  fractions,  since  the  numerator  and 
denominator,  of  each,  are  multiplied  by  the  same  number. 

a  b 

2.  Reduce    —    and    —    to  equivalent  fractions  having 

a  common  denominator. 

(■  the  new  numerators. 
bXb  =  b2  \ 

and  b  X  c  =  bc      the  common  denominator. 

55i  How  do  you  reduce  fractions  to  a  common  denominator  f 


ALGEBRAIC     FRACTIONS.  70 

Hence,  —  and  —  are  the  equivalent  fractions. 
be  be 

3.  Reduce  —    and    to  fractions  having  a  coin- 

b  c  s 

mon  denominator.  Ans.  —    and 


be  be 

Sx     25 

4.  Reduce      — -»     — >     and    d.    to   fractions   having    a 

2a      3c  '  s 

n  .  .        9cx      Aab  .    Gacd 

common  denominator.  Ans.  — — ?    - — >   and  • 

vac      vac  vac 

3       2x  2x 

5.  Reduce  —  >     — }    and    a  -\ ,  to  fractions   having 

a  common  denominator. 

9a       Sax  .    12a2  +  24a; 

AnS-   12a"'   T2T   and  12a       • 

1  o2  a2  -{-  a;2 

6.  Reduce     — ,     -rr->     aod     ?     to  fractions  hav- 

2  3  a  +  a; 

ing  a  common  denominator. 

3a  +  3a;     2a3  +  2a2a;  6a2  +  Qxz 

Ans'   6a~+6z    6a  +  6a;    '  and    6a  +  6a; ' 

£1  Az-f-j-  /tt2  ___  /r2 

7.  Reduce  —  >  -r— '     and     ; — >     to    a    common 

3o       5c  a 

denominator. 

.         hacd       l&abdx          ,      15a2bc  —  lbbcz7 
AnS'   ISbcH'     TEbcl>     and     15&crf " 

^    1-.   t  c       a  —  b  .  c 

8.  Reduce     — >     ■>    and     -■>    to  a   common 

oa         c  a  +  0 

denominator. 

.  ac2  +  c2^  5a3  —  5a62  n  5ac2 

-4»s.  — — — ,   -   ;  >     — — — ,   -     ->     and 


5a2c  +  5a6c      5a2c  -f-  5abc  5a2c  -f-  5a£c 


80 


ELEMENTARY     ALUEBRA. 


56.  To  add  fractional  quantities. 

EULE. 

Reduce  the  fractions,  if  necessary,  to  a  common  denomina- 
tor;  then  add  the  numerators  together,  and  place  their  sum 
over  the  common  denominator. 

EXAMPLES. 

1.  Add  §,  £,  and  §  together. 

By  reducing  to  a  common  denominator,  we  have 

6x3x5  =  90  1st  numerator. 

4  X  2  X  5  =  40  2d  numerator. 

2  X  3  X  2  =  12  3d  numerator. 

2  X  3  X  5  =  30  the  denominator. 

Hence,  the  expression  for  the  sum  of  the  fractions  becomes 

90       40       12  _  142 , 

30       30  +  30  ~"   30  ' 

which,  being  reduced  to  the  simplest  form  gives  4j-J. 

2.  Find  the  sum  of  — »    -=•>    and   —  • 

o      a  f 

Here     a  x  d  x  /  =  adf  -\ 

C  X  o  X  /  =  cbf   >  the  new  numerators. 

e  X  b  X  d  =  ebd  ) 
And      b  x  d  x  f  —  bdf     the  common  denominator. 

adf        cbf        ebd       adf  4-  cbf  4-  ebd      . 
Hence'  l^f  +  Mf  +  W  =  -J—bTf '  the  sum' 

56i  How  do  you  add  fractions  f 


ALGEBRAIC    FRACTIONS.  81 

2ax 

2abx  —  Scxz 


2.  To   a —    add    b  + 

o  c 


Ans.  a  +  6  + 


be 


4.  Add    — ,     —     and     —    together.  Ans.  +  rjr 

_     .  ,  _     x  —  2        .    4a;  .  .        19a;  —  14 

5.  Add     — - —  and    —   together.         Ans.    — — • 

3  7        &  21 

o     Mi  x-2       n         2x  -3       .        ."    ,  10a;-17 

6.  Add  x  -\ - —  to  3a;  -\ .  Ans.  4a;  +  - 


12 

7.  It  is  required  to  add  4a;,  -j—  >  and  — - —  together. 

.        .      ,  5a;3  -f  ax  +  a2 

Ans.  4x  -\ — ! • 

2aa; 

2a;    7a;  2a;  4-  1 

8.  It  is  required  to  add  — ,  —  >  and together. 

..        .      ,      49a; +12 
^2s+— go 

7a;  x 

9.  It  is  required  to  add  4a;,  — ,  and  2  +  —  together. 

44a;  +  90 

Ans.  4x  -) — " 

45 

2a;  8a; 

10.  It  is  required  to  add  3a;  +  —    and  x —   together. 

5  9 

23a; 
Ans.  3x  -f- 


45 

66 


11.  Required  the  sum  of  ac  —  —    and    1 r. 

*  8a  a 

8a2cd  —  Gbd  +  8ad  —  Sat 


Ans 


8ad 
4* 


82 


ELEMENTARY  ALGEBRA. 


CASE  VI. 

57    To  subtract  one  fractional  quantity  from  another. 

RULE. 

I.  Reduce  the  fractions  to  a  common  denominator. 

II.  Subtract  the  numerator  of  the  subtrahend  from  the 
numerator  of  the  minuend,  and  place  the  difference  over  the 
common  denominator. 

EXAMPLES. 

3  2 

1.  What  is  the  difference  between   — -  and  — -• 

7  8 

_3_        _2     _  24  _  14  _  10        Ji     j 

x  '     a  2<x  —-■  4# 

2.  Find  the  difference  of  the  fractions  ■   and — • 

2o  3c 

,    (x  —    a)  x  3c  =  ocx  —  3ac  )    . 
Here, K/^  .  \       ^,        .  ,       „,     i  the  numerators. 


(2a  —  4s:)  X  26  =  4a6  —  86^ 

And,  26  X  3c  =  66c  the  common  denominator. 

TT          3cx— Sac       4ab—Sbx       Scx—3ac—4ab-\-8bx      . 
IIenCe'-(^ 6bc~  ~ Qbc •  Am' 

3.  Required  the  difference  of  — —  and  —  •        Ans.  ■  —  • 

4.  Required  the  difference  of  5y  and    ■—■.       Ans.     -—■• 

8  8 

_,       .     ,   .      ,._  .3a;  2x  ISx 

5.  Required  the  difference  of   —  and  —  •       Ans.———' 


67-  How  do  you  subtract  fractious  ? 


ALGEBRAIC     FRACTIONS.  83 

x  ~4~  (t  c 

6.  Required  the  difference  between    — ; —  and  — . 

b  a 

.       dx  +  ad  —  be 

Ans- Yd 

7.  Required  the  difference  of  — — —    and    — - — . 

56  8 

24a?  +  8a  —  IO&b  — 355 

Ans.  — ; . 

406 

X  X  —  CL 

8.  Required  the  difference  of   3x  +  -r-     and  a; . 

be 

a       r.     ,   cx  +  bx  —  ab 

Ans.  2x  H ; . 

be 

CASE   VII. 

58.  To  multiply  fractional  quantities  together. 

RULE. 
If  the  quantities  to  be  multiplied  are  mixed,  reduce  them  to 
fractional  forms  ;  then 'multiply  the  numerators  together  for 
a  numerator  and  the  denominators  together  for  a  denominator. 


1.  Multiply  -    of    -    by    8} 


EXAMPLES. 

1       .     3 

6    °f     7 

Operation. 

We  first  reduce  the  com-  13         3 

pound  fraction  to  the  sim-  6  7  —  42 ' 

pie  one  ^,  and  then   the  o^ 

mixed  number  to  the  equiv-  §\  =  — 

alent    fraction     2^  ;    after 


which,    we     multiply    the     „  3 

numerators  and  denomina-  '     42       3        12G      42 


25  _  75  _  25 

~W  ~  126  ~  42 

tors  together.  A       25 

fo  Ans.  — . 

42 


»4  ELEMENTARY     ALOEBltA. 

^    -.r  -i  .  -i  hx  ,       c        ^.  bx       a2  +  bz 

2.  Multiply   a  +  —  by  —  •     1  irst,   a  -j = • 

*  "*  ad  a  a 

TT  a2  +  &c  c  a2c  +  &c#         . 

Hence,  X   — r  = 5 ■     -4ws. 

a  a  aa 

n    „       .     n    ,  ,  „  3a;       ,  3a  .         9ax 

3.  Required  the  product  01    —  and  —  •  Ans.  -rr« 


2x  3x2 

4.  Required  the  product  of   -—   and  — — 


3z3 
Ans.    ——  • 
ba 

^.    ,    ,  .  -,  <.  2x    3ab         ,  3ac 

5.  Find  the  continued  product  ol   — >    and  — -  • 

r  a        c  26 

-4«5.  9a£. 

bx  a 

6.  It  is  required  to  find  the  product  of  0  H and  —  • 

.         ab  +  bx 

Ans.    

x 

x2  —  b2  x2  +  b2 

7.  Required  the  product  of  — 7 and      , 

x*  —  b* 
Ans. 


b2c  +  be2 

X  -\-  I  X  —  1 

8.  Required  the  product  of  x  H >  and  — ——. -  • 

ax2  —  ax  +  x2  —  1 


Ans. 


a2  +  ab 

ax  n2  —  f 

9.  Required  the  product  of  oH by 

Ans. 


x  +  x2 
a4  —  a2x2 


ax  +  ax2  —  x2  —  x'6 


68.  How  do  you  multiply  fractions  together  ? 


ALGEBRAIC     FRACTIONS.  85 

CASE     VIII. 

59.  To   divide   one  fractional   quantity  by  another. 

EULE. 

Reduce  the  mixed  quantities,  if  any,  to  fractional  forma  ; 
then  invert  the  terms  of  the  divisor  and  multiply  as  in  the  last 
case. 

EXAMPLES. 

1.  Divide .     .     .     .  —    by   —  • 

24      '     8 

The  true  quotient  will  be   expressed  by  the  complex 

10 

fraction  —• 
¥ 
Let  the  terms  of  this  fraction  be  now  multiplied  by  the 

denominator  with  its  terms  inverted :  this  will  not  alter  the 

value  of  the  fraction ;  and  we  shall  then  have, 


10         io  Y  i        1A 

24    24    ■*■    5    24 

8  8      A    5 


j-^  =  J$  *  !  =  t  =  quotient. 


It  will  be  seen  that  the  quotient  is  obtained  by  simply 
multiplying  the  numerator  by  the  denominator  with  its 
terms  inverted.  This  quotient  may  be  further  simplified  by 
dividing  by  the  common  factors  5  and  8,  giving  §  for  the 
true  quotient. 

h  f 

2.  Divide  .     .     a  —  —    by   —  • 

2c      J     g 

b        2ac  —  b 


2c  ~       2c 

„  b         f        2ac  —  b        a        2acg  —  bg         . 

Hence,  a  — —  —  —  — x  4-  =  — t  ,       •     ^ns 

2c        g  2c       A  /  2c/ 

69t  IIow  do  you  divide  one  fraction  by  another? 


86  ELEMENTARY     ALGEBRA. 

7*    ,      a.  .,   ,,        12  .         91a? 

3.  Let     —     be  divided  by     —  •  Ans.   -rjr- 

4.^2  ^.^ 

4.  Let     — -—     be  divided  by     5x.  Ans.  — • 

7  J  35 

-_         ji  +  1     ,       ,..,,,         2a;  .         a;  +  1 

5.  Let     — - —     be  divided  by     —  • 

6  J       3 

x  x 

6.  Let be  divided  by     —  • 

x  —  1  J      2 

7.  Let    — -    be  divided  by     —  • 
8'   Let      ^T      b6  diYided  hj     |f 

°-Let   ^^T^    bedividedb?    T^i 

-4ns.  x  H • 

x 

10.  Divide    Ga2  +  -|-   by   c2  -   a 


4a 

Ans. 

2 
a:  — 1 

Ans 

55a; 
'  ""2tT' 

Ans.    ■ 

2;  —  b 
Cc2.r 

x2  +  bx 

5       J  2 

GOa2  +  25 


u4«s. 


10c2  —  bx  +  5a 


11.  Divide   18c2  -  x  +  ~    by    a2  -  A 
6       J  5 


906c2  —  55a;  +  5a 


5a26  —  J2 


12.  Divide   20a;2  •-  -^    by  a;2  - C. 


20cfc3/f2  -  8abf 
"*"  dcP/x2  -  dc^b  +  dc6' 


EQUATIONS     Ol"     1UK     Fill  ST     DKUliKE.  87 


CHAPTER  III. 

Of  Equations  of  the  First  Degree. 


60.  An  Equation  is  the  algebraic  expression  of  two  equa) 
quantities  with  the  sign  of  equality  placed  between  them. 
Tnus,.r  =  a  +  b  is  an  equation,  in  which  x  is  equal  to  the 
sum  of  a  and  b. 

61.  By  the  definition,  every  equation  is  composed  of  two 
parts,  connected  by  the  sign  =  .  The  part  on  the  left  of  the 
sign,  is  called  the  first  member  ;  and  that  on  the  right,  the 
second  member.  Each  member  may  be  composed  of  one  or 
more  terms.  Thus,  in  the  equation  x  =  a  -\-  b,  x  is  the  first 
member,  and  a  +  b  the  second. 

62.  Every  equation  may  be  regarded  as  the  algebraic 
enunciation  of  some  proposition.  Thus,  the  equation 
x  +  x  =:  30,  is  the  algebraic  enunciation  of  the  following 
proposition : 

6d  What  is  an  equation  ? 

CI.  Of  how  many  parts  is  every  equation  composed?  Hew  are  the 
parts  connected  with  each  other  ?  What  is  the  part  on  the  left  called  > 
What  is  the  part  on  the  right  called  ?  May  each  member  be  composed 
of  one  or  more  terms  ?  In  the  equation  x  —  a  +  b,  which  is  the  first 
member  ?  Which  the  second  ?  How  many  terms  in  the  first  member  f 
How  uumv  in  the  "ecoud  ? 


88  ELEMENTARY     ALGEBRA. 

To  find  a  number  which  being  added  to  itself,  shall  give  a 
sum  equal  to  30. 

Were  it  required  to  solve  this  problem,  we  should  first 
express  it  in  algebraic  language,  which  would  give  the 
equation 

x  +  x  =  30. 
By  adding  x  to  itself,  we  have  9 

2x  =  30. 
And  by  dividing  by  2,  we  obtain 
x=  15. 

Hence,  we  see  that  the  solution  of  a  problem,  by  algebra, 
consists  of  two  distinct  parts  :  viz.  the  statement  of  the 
problem,  and  the  solution  of  an  equation. 

I.  The  statement  consists  in  expressing  algebraically  the 
relation  between  the  known  and  the  required  quantities. 

II.  The  solution  of  the  equation  consists  in  finding  the 
values  of  the  required  quantities  in  terms  of  those  which  are 
known. 

The  given  or  known  parts  of  a  problem,  are  represented 
either  by  figures  or  by  the  first  letters  of  the  alphabet,  a,  6, 
c,  &c.  The  required  or  unknown  parts  are  represented  by 
the  final  letters,  x,  y,  z,  &c. 


Find  a  number  which,  being  added  to  twice  itself,  the 
sum  shall  be  equal  to  24. 

62i  How  may  you  regard  every  equation  ?  What  proposition  does 
the  equation  x  +  x  —  30  state  ?  Of  how  many  parts  does  the  solution 
of  a  problem  by  algebra,  consist?  Name  them.  In  what  does  the  1st 
part  consist  ?  What  is  the  2d  part  ?  By  -what  are  the  known  parts  of 
a  proposition  represented  !  By  what  are  the  unknown  part9  represented  ? 


EQUATIONS     OF     THIS     FIRST     UEGRKfi.  89 

Statement. 
Let  x  denote  the  number.     We  shall  then  have 

x  +  2x  —  24. 
This  is  the  statement. 

Solution. 

Having     .     .     .     .  x  -{-  2x  =  24, 

we  add x  -f-  2.t, 

which  gives       ...  3x  =  24, 

and  dividing  by  3,      .  $  =  8. 

63.  The  value  found  for  the  unknown  quantity  is  said  to 
be  verified,  when,  being  substituted  for  it,  in  the  given  equa- 
tion, the  two  members  are  proved  equal,  each  to  each. 

Thus,  in  the  last  equation  we  found  x  =  8.  If  we  substi- 
tute this  value  of  x  in  the  equation 

x  +  2sc—  24, 
we  shall  have      8  +  2  X  8  =  8  +  16  =  24. 
which  proves  that  8  is  the  true  answer. 

64.  An  equation  involving  only  the  first  power  of  the 
unknown  quantity,  is  called  an  equation  of  the  first  degree. 
Thus,  Qx  -f-  Sx  —  5  =  13, 

and  ax  +  bx  -f-  c  =  d, 

are  equations  of  the  first  degree. 

By  considering  the  nature  of  an  equation,  we  see  that  it 
must  possess  the  three  following  properties  : 

(j<3.  When  is  an  equation  said  to  be  verified? 

64t  When  an  equation  involves  only  the  first  power  of  the  unknown 
quantity,  what  is  it  called  ?  What  are  the  three  essential  properties  of 
every  equation  ? 

5 


00  KLKMENTAXiY     ALGKBIIA 

1st.  The  two  members  must  be  composed  of  quantities  of 
the  same  k.nd  :  that  is.  dollars  =  dollars,  pounds  =  pounds. 
2d.  The  two  members  must  be  equal  to  each  other. 
3d.  The  two  members  must  have  like  signs. 

65.  An  axiom  is  a  self-evident  truth.  We  may  lere 
state  the  following. 

1.  If  equal  quantities  be  added  to  both  members  of  a>i  cqua* 
tion,  the  equality  of  the  members  tuill  not  be  destroyed. 

2.  If  equal  quantities  be  subtracted  fr^rti  both  members  of 
an  equation,  the  equality  ivill  not  be  destroyed. 

3.  If  both  members  of  an  equation  be  multiplied  by  the  same 
number ,  the  equality  ivill  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same 
number,  the  equality  will  not  be  destroyed. 

Transformation  of  Equations. 

66.  The  transformation  of  an  equation  consists  in  chang 
ing  its  form  without  affecting  the  equality  of  its  members. 

The  following  transformations  are  of  continual  use  in  the 
resolution  of  equations. 

First  Transformation 

67.  When  some  of  the  terms  of  an  equatior  are  frac- 
tional, to  reduce  the  equation  to  one  in  which  the  terms 
shall  be  entire. 

1.  Take  the  equation 

2x      ox        x 

o  4  O 

65.  What  is  an  axiom  ?     Name  the  four  axioms  ? 

CG.  What  is  the  transformation  of  an  equation  ? 

G7.  What  is  the  first  transformation  ?  What  is  the  least  •^mmon 
multiple  of  several  numbers  ?  How  do  you  find  the  lenst  coiumob 
multiple ! 


EQUATIONS     OF     THE     FIRST     UEUUEE.  9] 

First,  reduce  all  the  fractions  to  the  same  denominator, 
by  the  given  rule ;  the  equation  then  becomes 

48a;       54a:        12a;  _ 
"72"  ~~  "72"  +  ~W  ~       ' 

and  since  we  can  multiply  both  members  by  the  same  mim 
ber  without  destroying  the  equality,  we  will  multiply  them 
by  72,  which  is  the  same  as  suppressing  the  denominator 
72,  in  the  fractional  terms,  and  multiplying  the  entire  term 
by  72 ;  the  equation  then  becomes 

48a;  —  54a;  +  12a;  =  792, 
or  dividing  by  6,    8a;  —    9x  +    2x  =  132. 

But  this  last  equation  can  be  obtained  in  a  shorter  way,  by 
finding  the  least  common  multiple  of  the  denominators. 

The  least  common  multiple  of  several  numbers  is  the  least 
number  which  they  will  separately  divide  without  a  remainder. 
When  the  numbers  are  small,  it  may  at  once  be  determined 
by  inspection.  The  manner  of  finding  the  least  common 
multiple  is  fully  shown  in  Arithmetic  §  87. 

Take  for  example,  the  last  equation 

¥~"T  +  "6=     " 

We  see  that  12  is  the  least  common  multiple  of  the  de- 
nominators, and  if  we  multiply  all  the  terms  of  the  equa- 
tion by  12,  we  obtain 

8a;  —  9a:  +  2a;  =  132  ; 

tJio  same  equation  as  before  uniud. 


92  ELEMENT  AKY      A  L  O  E  B  H  A  . 

68.  Hence,  to  make  the  denominators  disappear  from  an 
equation,  we  have  the  following 

RULE. 

L  Find  the  least  common  multiple  of  all  the  denomina- 
tors. 

II.  Multiply  every  term  of  both  members  of  the  equation  by 
this  common  multiple — reducing  at  the  same  time  the  frac- 
tional to  entire  terms. 

EXAMPLES. 


1.  Clear  the  equation   —  +  —  —4  =  3     of  its  denomi- 
nators. Ans.  7x  +  5x  —  140  =  105. 

2.  Clear  the  equation   —  +  —  —  —  =  8     of  its  denom- 

0        9       27 

inators.  Ans.  dx  +  Gx  —  2x  =  432. 

X  X  X  X 

3.  Clear  the  equation    —  +  —  —  — -  +  —  =  20      of  its 

£  O  o  x4f 

denominators.  Ans.  18*  +  12a  —  4x  +  Sx  =  720. 

xxx 

4.  Clear  the  equation    —  +  —  —  —  =  4     of  its  denom- 

inators.  Ans  14z  +  lOx  —  35jt  =  280. 

5.  Clear  the  equation !-  — -  =  15     of  its  denom- 

^  4         5        6 

toators.  Ans.  \f>x  —  12x  +  10.r  =  900. 


C8.  Give  tlic  mlo  for  clearing  an  equation  ol  its  denominators. 


LIQUATIONS     OF     T  n  E     FIKST     DEGREE.  98 

31  QC  +C  3/ 

6.  Clear  the  equation    — —  +  —  +  -7-  =  12    of  its 

4         o  o  9 

denominators.  _4»s.  18a;  —  12z  +  9x  +  8z  =  8G4. 

CL  C 

7.  Clear  the  equation    — —  +  /  =  g. 

Ans.  ad  —  be  +  bdf  •=.  bdg. 

8.  In  the  equation 

ax       2c2x  4ibc2x       5a3    ,    2c2 

+  4a  =  —3-  _  —  + 36, 

b  ab  ■  a3  bl  a 

the  least  common    multiple  of  the   denominators  is  a3b2 ; 

hence,  clearing  the  equation  of  fractions,  we  obtain 

a*bx  —  2a2bc2x  +  4a*b2  =  U3c2x  —  5a6  +  2a2b2c2  —  3a3b3. 


Second  Transformation. 

69.  When  the  two  members  of  an  equation  are  entire 
polynomials,  to  transpose  certain  terms  from  one  member 
to  the  other. 

1.  Take  for  example  the  equation 

5x  -  6  =  8  +  2x. 

If,  in  the  first  place,  we  subtract  2x  from  both  members, 
the  equality  will  not  be  destroyed,  and  we  have 

5x  —  6  —  2x  =  8. 

Whence  we  see,  that  the  term  2x,  which  was  additive  in 
the  second  member,  becomes  subtractive  by  passing  into 
the  first. 


69.  What  is  the  second  transformation?  What  do  you  understand 
by  transposing  a  term  ?  Give  the  rule  for  transposing  from  one  member 
to  the  other. 


94  ELEMENTARY      ALGEBRA. 

In  the  second  place,  if  we  add  6  to  both  members,  the 
equality  will  still  exist,  and  we  have 

5x  —  6  —  2x  +  G  =  8  +  G. 

Or,  since   —  6  and   +  6  destroy  each  other,  we  have 

5x  —  2x  —  8  +  G. 

Hence  the  term  which  was  subtractive  in  the  first  mem 
ber,  passes  into  the  second  member  with  the  sign  of 
addition. 

2.  Again,  take  the  equation 

ax  +  b  =z  d  —  ex. 

If  we  add  car,  to  both  members,  and  subtract  b  from 
each,  the  equation  becomes 

ax  +  b  +  ex  —  b  =  d  —  ex  -j-  ex  —  b. 

or  reducing  ax  +  ex  =  d  —  b. 

When  a  term  is  taken  from  one  member  of  an  equation 
and  placed  in  the  other,  it  is  said  to  be  transposed. 

Therefore,  for  the  transposition  of  the  terms,  we  have  the 
following 

RULE. 

Any  term  of  an  equation  may  be  transposed  by  simply 
changing  its  sign  from  -f-  to  — ,  or  from  —  to  -f-. 

70.  We  will  now  apply  the  preceding  principles  to  the 
resolution  of  equations. 

1.  Take  the  equation 

4*  —  3  =  2x  4-  5. 


EQUATIONS     OF     TUE     F  I  It  S  T     D  E  O  It  E  K  .  t>5 

By  transposing  the  terms  —  3  and  2#,  it  becomes 

Ax  —  2x  =  5  +  3. 

Or,  reducing  2x  =  8. 

8 
Dividing  by  2  x  =  —  =  4. 

Verification, 

If  now,  4  be  substituted  in  the  place  of  x,  in  the  given 
equation 

4.x  —  3  =  2x  +  5, 
it>  becomes  4x4  —  8  =  2x4  +  5. 

or,  13  =  13. 

Hence,  the  value  of  x  is  verified  by  substituting  it  for  the 
unknown  quantity  in  the  given  equation. 

2.  For  a  second  example,  take  the  equation 

5x       Ax  __  7        \ox 

l2~~  ll~    °  ~~8  G~" 

By  causing  the  denominators  to  disappear,  we  have 

10a;  —  o2x  —  312  =  21  —  52x, 
or,  by  transposing 

10*  -  S2x  +  52a;  =  21  +  312 
try  reducing  30a;  =  333 

333        111 

&  result  which  may  be  verified  by  substituting  it  for  x  in  the 
given  equation. 

3.  For  a  third  example  let  us  take  the  equation 

(3a  —  x)  (a  —  b)  -f-  2ax  =  46  (x  -f  a). 


06  ELEMENTARY     ALGEBRA. 

It  is  first  necessary  to  perform  the  multiplications  indicat- 
ed, in  order  to  reduce  the  two  members  to  polynomials. 
This  step  is  necessary  before  we  can  disengage  the  unknown 
quantity  x,  from  the  known  quantities.  Having  done  that, 
the  equation  becomes, 

3a2  —  ax  —  Sab  -f-  bx  +  Zaz  =  AJbx  -\-  4a6, 
or,  by  transposing 

—  ax  -f  bx  +  2aa;  —  4bx  =  4ab  +  3a6  —  3a3, 
by  reducing  ax  —  Sbx  ==  7«5  —  3a2; 

Or,  (Art.  41).  (a  -  Sb)x  =  7a6  -  3a2. 

Dividing  both  members  by  a  —  36  we  find 
_  lab  -  3a2 
X~      a -36  ' 
Hence,  in  order  to  resolve  an  equation  of  the  first  degree, 
we  have  the  following 

UTILE. 

I.  If  there  are  any  denominators,  cause  them  to  disappear, 
and  perform,  in  both  members,  all  the  algebraic  operations 
indicated. 

II.  Then  transpose  all  the  terms  containing  the  unknown 
quantity  into  the  first  member,  and  all  the  knoivn  terms  into 
the  second  member. 

III.  Reduce  to  a  single  term  all  the  terms  involving  the  un 
known  quantity  :  this  term  ivill  be  composed  of  two  factors, 
one  of  which  will  be  the  unknown  quantity,  and  the  other  its 
multipliers,  connected  by  their  respective  signs. 

IV.  Divide  both  members  of  the  equation  by  the  multiplier 
of  the  unknown  quantity. 

TO.  What  is  the  first  step  in  resolving  an  equation  of  the  first  degree  I 
What  the  second  ?     What  the  third  ?     What  the  fourth  ( 


EQUATIONS     OF    THE     FIRST     DEUKEE.  9' 


EXAMPLES. 

1.  Given  3a  —  2  +  24  =  31    to  find  x.  Ans.  x  =  3. 

2.  Given  x  -f-  18  =  3x  —  5   to  find  x.      Ans.  x  =  11  — . 

3.  Given   6  —  2x  -f  10  =  20  —  3x  —  2  to  find  x. 

Ans.  x  =  2. 

4.  Given  x-\-— x  +  —  x  =  ll    to  find  x.     Ans.  x  =  6 

1  6 

5.  Given  2x  — —  x  -\- 1  =5x  —  2,  to  find  x.  Ans.  #=-=r» 

6.  Given   Sax  +  — 3  =  hx  — -  a,  to  find  x. 

6  — 3a 
AnS'X  =  Ga^2b' 

7.  Given  ^=^  +  f.  =  20  -  ^=-H_to  find  ar. 

.4ns.  a;  =  23  -T- 
4 

„    -.         *  +  3  ,    *        .      x  —  5  x    .    , 

8.  Given  — s \-  —  =  4 ; —  to  find  a*. 

2  3  4 

f» 

Ans.  x  =  3— 
Id 

x       3a?               4a? 
0.  Given   - —  ~-  +  x  =  — 3   to  find  x. 

4        Z  o 


.4ns.  a?  =  4. 


10.  Given ; 4  =  /  to  find  a?. 

e  a 


_cd£+4cd 
Ant'  z  ~  Ud-2bc 


98  ELEMEHTABT     ALGEBRA. 


_,.        8ax  —  6       36  — c 

11.  Given  = —  =4  —  6    to  find    x. 


56  +  96  -  7e 
Ans.  x  = 


16a 


^          x        x  —  2       x       13 
12.  Given  — (-  -^  —  —   to  ^^    a;* 

O  o  m  o 


„4ns.  a:  =  10. 


*        x        a;         a; 

13.  Given rH =f    to  find     a;. 

abed 

.  abedf 

Ans.  x  — 


bed  —  acd  +  abd  —  abc 
Note. — What  is  the  numerical  value  of  a:,  when  a  =  1, 
ft  =  2,  c  =  0,  rf  -=  4,  6  :    h,  and  /  =  6. 

14.  Given  1  -  ~  -  £-=i  =  -  12||     to  find    x. 

Ans.  x  =  14. 

..    _.  3a:  —  5       4a;  —  2  .  . 

15.  Given  a? — 1 — —  =  x  -f-  1     to  find    x. 

lo  1 1 

-4ns.  x  =  6. 

16.  Given  a;+  —  +  -r  —  4=2a;  —  43     to  find    ar. 

4        5         6 

^4?w.  a?  =  60. 


, ■  4a;  -  2      3a;  -  1 

f.  Given  2a; —  =  — - —     to  find    x. 


Ans.  x  =  3. 


¥  7 

18.  Given  3a;  ^ - —  =  x  +  a     to  find     x. 

3a +  d 

Ans.  x  =  • 

6  +  6 

,ft   n-         oa»  — b  ,.  a       6a;       62;— a 

19.  Given  — - H  —  =  — - —    to  find    x. 

4  o  m  o 

Ans.  x  =  ,- 


3a-26 


EQUATIONS    0¥    THE     FIRST    DEGREE.  99 

20.  Find  the  value  of  a;  in  the  equation 

(a  -M)   (x  -  ft)  4«6  -  h2  a?  —  hx 

a  —  o  a  -J-  o  o 

«4  -f  3a3ft  -f  4a262  —  Gaft3  4-  2ft4 

,4»$.    *  =  7 ^   r 

2i(^a2  +  a»  —  J  ) 

0/"  Propositions  giving   rise  to  Equations  of  the   First 
Degree  involving  but  one  unknown  quantity. 

71.  It  has  already  been  observed  (Art.  62),  that  the 
solution  of  a  problem  by  algebra,  consists  of  two  distinct 
parts  : 

1st.  To  make  the  statement  :  that  is,  to  express  the  con- 
ditions of  the  proposition  algebraically  ; 

2d.  To  solve  the  resulting  equation  :  that  is,  to  disengage 
the  known  from  the  unknown  quantities. 

We  have  already  explained  the  manner  of  finding  the 
value  of  the  unknown  quantity,  after  the  proposition  has 
been  stated.  It  only  remains  to  point  out  the  best  methods 
of  stating  the  proposition  in  the  language  of  algebra. 

This  part  of  the  algebraic  solution  of  a  problem  cannot, 
like  the  second,  be  subjected  to  any  well  defined  rule. 
Sometimes  the  enunciation  of  the  proposition  furnishes  the 
equation  immediately  ;  but  sometimes  it  is  necessary  to 
discover,  from  the  enunciation,  new  conditions  from  which 
an  equation  may  be  deduced. 

71.  Into  how  many  parts  is  the  resolution  of  a  problem  in  algebra 
divided  ?  What  is  the  first  step  ?  "What  the  second  \  Which  part  lias 
already  been  explained?  Which  part  is  now  to  be  considered?  Can 
this  part  be  subjected  to  exact  rules  ?  Give  the  general  rule  for  slating 
*  proposition. 


100  ELEMENTARY     AL&3BKA. 

In  almost  all  cases,  however,  we  are  able  to  make  the 
statement ;  that  is,  to  discover  the  equation,  by  applying 
the  following 

KULE. 

Represent  the  unknown  quantity  by  one  of  the  final  letters 
of  the  alphabet ;  and  then  indicate  by  means  of  the  algebraic 
signs,  the  same  operations  on  the  known  and  unknown  quan 
tities,  as  would  verify  the  value  of  the  unknown  quantity, 
were  such  value  known. 

QUESTIONS. 

1.  To  find  a  number  to  which  if  5  be  added,  the  sum  wili 
be  equal  to  9. 

Denote  the  number  by     m. 
Then  by  the  conditions 

x  -f  5  =  9. 
This  is  the  statement  of  the  proposition. 
To  find  the  value  of  x,  we  transpose  5  to  the  second 
member,  which  gives 

2  =  9-5=4. 

Verification. 
4  +  5  =  9. 

2.  Find  a  number  such,  that  the  sum  of  one-half,  one- 
third,  and  one-fourth  of  it,  augmented  by  45,  shall  be  equal 
to  448. 

Let  the  required  number  be  denoted  by     x. 

Then  one-half  of  it  will  be  denoted  by 


x 

a" 

X 

¥ 

one-fourth          M  "  by     — 

4 


one-third  "  "  by 


EQUATIONS     OS"    TfiE    FIRST     DEGREE.        101 

And  by  the  conditions, 

XXX 

2"f"3  +  4+45  =  448* 
This  is  the  statement  of  the  proposition. 
To  find  the  value  of  x,  subtract  45  from  both  membeis, 
this  gives 

At  /)••  A* 

_  +  _+_=403. 

By  clearing  the  equation  of  denominators,  we  obtain 

Gx  +  4x  +  3x  =  4836, 
or  13z  =  4836. 

Hence,  x  =  — —  =  372. 

lo 

<>70         Q70        Q70 

^  +  211  +  ^f-  +  45  =  186  +  124  4-  93  4-  45=448, 
2  o  4 

3.  What  number  is  that  whose  third  part  exceeds  its 
fourth  by  16  % 

Let  the  required  number  be  represented  by  x.     Then, 

-—  x  =     the  third  part. 

o 

~—  x  =     the  fourth  part. 

And  from  the  conditions  of  the  problem 

1  1 

—  x -x=lQ. 

3  4 

This  is  the  statement.     To  find  the  value  of  x,  we  clea? 

the  equation  of  the  denominators,  which  gives 

4x  —  3x=  192. 

and  x  =  192. 


102  ELEMSNTART     ALGEBRA. 

Verification. 

15i_^  =  04-48  =  10. 
3  4 

4.  Divide  $1000  between  A,  B  and  C,  so  that  A  shall 
have  $72  more  than  B,  and  C  $100  more  than  A. 

Let  x  =  B's  share  of  the  $1000. 

Then  x  +    72  =     A's  share, 

and  x  +  172  =     C's  share, 

their  sum  is         Zx  4-  244  =$1000. 

This  is  the  statement. 

By  transposing  244  we  have 

Zx  =  1 000  —  244  =  756 

and  x  =  -——  —  252  =  B's  share. 

o 

Hence,  x  4-    72  =  252  4-    72  =  $324  =     A's  share. 

And  x  +  172  =  252  +  172  =  $424  =     C's  share. 

Verification. 
252  4-  324  4-  424  =  1000. 

5.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third 
part,  21  gallons  were  afterwards  drawn,  and  the  cask  being 
then  guaged,  appeared  to  be  half  full  :  how  much  did  it 
hold] 

Suppose  the  cask  to  have  held     x     gallons. 

x 
Then,         —    what  leaked  away. 

o 

x 
And  —  4-  21  =    what  had  leaked  and  been  drawn. 

o 

x  x 

Hence,      —  4-  21  =  —    by  the  conditions. 
3  2      J 

This  is  tli©  statement 


EQUATIONS     OF     THE     FIRST     DEGREE.        103 

To  find  x,  we  have 

2x  +  126  =  3x, 
and  —    x    =  —  126, 

and  by  changing  the  signs  of  both  members,  which  does  not 
destroy  their  equality,  we  have 

*  =  126. 

Verification. 

J2»  +  »,-=  42  +  2i  =  63  =  !§1 

6.  A  fish  was  caught  whose  tail  weighed  9$.,  his  head 
weighed  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighed  as  much  as  his  head  and  tail  together ;  what  was 
the  weight  of  the  fish  1 

Let  2x  =  the  weight  of  the  body. 

Then,  9  +  x  =  weight  of  the  head  ; 

and  since  the  body  weighed  as  much  as  both  head  and  tail, 

2x  =  9  +  9  +  x, 

which  is  the  statement.     Then, 

2x  —  x  =  18     and     x  =  18. 

Hence  we  have, 

2x  =  SGlb.  «=  weight  of  the  body. 
9  +  x  =  27 lb.  =  weight  of  the  head. 
9lb.  =  weight  of  the  tail. 


Ilenoo,  72/6.  =  weight  of  the  fish. 


104  ELKMEKTARY     ALGEBRA, 

7.  The  sum   of  two  numbers  is  67  and  their  difference 
19  :  what  are  the  two  numbers  ? 

Let  x  =  the  less  number. 

Then,  x  +  19  =  the  greater. 

and  by  the  conditions 

2x  +  19  =  67. 

This  is  the  statement. 

To  find  x,  we  first  transpose  19,  which  gives 

2x  =  67  -  19  =  48  ; 

48 
hence,        x  =  —  =  24,    and    x  -f  19  =  43. 

Verification. 
43  +  24=67,    and    43-24  =  19. 

Another  Solution. 
Let  x  denote  the  greater  number  : 
then  x  —  19  will  represent  the  less, 

and,  2x  —  19  =  67,  whence  2x  =  67  +  19  j 

therefore,  x  =  —  =  43, 

and  consequently  x  —  19  =  43  —  19  =  24. 

General  Solution  of  this  Problem. 

The  sum  of  two  numbers  is   a,  their  differo-not.  b    b 
What  are  the  two  numbers  ? 


EQUATIONS     OF     THE     FIRST     DEGREE.         105 

Let  x  denote  the  least  number. 

Then,  x  +  b  will  represent  the  greater. 

Hence,         2x  -\-  b  =  a,      whence      2x  =  a  —  b  ; 

x,        -  a  —  b         a         b 

therefore,  x  =  — - —  :=  —  —  —  , 

a         b    ,    ,         a    ,    b 
and  consequently,    a;  -f-  o  =  —  —  —  +  6  =  — -4-—.. 

As  the  form  of  these  two  results  is  independent  of  the 
values  attributed  to  the  letters  a  and  b,  it  follows  that, 

Knowing  the  sum  and  difference  of  two  numbers,  we  obtain 
the  greater  by  adding  the  half  difference  to  the  half  sum,  an? 
the  less,  by  subtracting  the  half  difference  from  half  the  sum. 

Thus,  if  the  given  sum  were  237,  and  the  difference  91! 

.     237       99          237  +  99      33G 
the  greater  is    —  -f-   or =  —  =  168  j 

,  4l     .  237       99  138 

and  the  least    — — ,  or  ——  =  69. 

<&  4>  2 


Verification. 
1G8  +  69  =  237    and    168  -  69  =  99. 

8.  A  person  engaged  a  workman  for  48  days.  For  each 
day  that  he  labored  he  received  24  cents,  and  for  each  day 
that  he  was  idle,  he  paid  12  cents  for  his  board.  At  the 
end  of  the  48  days,  the  account  was  settled,  when  the  laborel 
received  504  cents.  Required  the  number  of  working  days, 
and  the  number  of  days  he  was  idle. 

5* 


10(1  ELEMENTARY     ALGEBRA, 

V 

If  the  two  numbers  were  known,  and  the  first  multiplied 
by  24,  and  the  second  by  12,  the  difference  of  these  pro- 
ducts would  be  504.  Let  us  indicate  these  operations  by 
means  of  algebraic  signs. 

Let  x  =   the  number  of  working  days 

48  —  x  =  the  number  of  idle  days 
Then      24  X  x  =  the  amount  earned 
and   12(48  —  x)=  the  amount  paid  for  board. 
Then,    24.r  -  12(48  -  x)  =  504 
what  was  received,  which  is  the  statement. 
Then  to  find  x  we  first  multiply  by  12,  which  gives 

24.C-576  +  12.r  =  504. 
or,  SGx  =  504  +  570  =  10S0, 

and  x  —    -  -    =z  30  the  number  of  working  days: 

whence,  4S  —  30  =  18      the  number  of  idle  days. 

Verification, 

.  Thirty   days'   labor,   at  24  cents 

a  day,  amounts  to 30  X  24  =  720  cents. 

And  18  day's  board,  at  12  cents 
a  day,  amounts  to 18  X  12  =  210  cents. 

The  difference  is  the  amount  received  504  cents. 

General  Solution. 

This    problem    may  be   made  general,  by  denoting  the 
whole  number  of  working  and  idle  days  by  n. 
The  amount  received  for  each  day's  work  by  a. 
The  amount  paid  for  board,  for  each  idle  day,  by  b. 


EQUATIONS     OF     THE     FIRST     DEGREE.         107 

And  the  balance  due  the  laborer,  or  the  result  of  the 
account,  by  c. 

As  before,  let  the  number  of  working  days  be  denoted' 
by  x. 

The  number  of  idle  days  will  then  be  expressed  by  n — x. 

Hence,  what  is  earned  will  be  expressed  by  ax. 

And  the  sum  to  be  deducted,  on  account  of  board,  by 
b(n  —  x). 

The  statement  of  the  problem,  therefore,  is 

ax  —  b{n  —  x)  =  c. 

To  find  x,  we  first  multiply  by  b,  which  gives 

ax  —  bn  -f-  bx  =  c, 

or,  (a  +  b)x  =  c  -}-  bn, 

c  +  bn  .         .        ,  .       _ 

wnence,  x  =  —  =     number  of  working  days. 

,  ,  c  -f-  bn      an  -\-  bn  —  c  —  bn 

and  consequently,  n  —  x  =  n —  = ; — ; , 

x  a  +  6  a-f-6 

->r,  n  —  x  = —  =     number  of  idle  days. 

a  +  b  J 

Let  us  now  suppose  n  =48,  a  =  24,  b  =  12,  and  c  •=  504. 
These  numbers  will  give  for  x  the  same  value  as  before 
found. 

0.  A  person  dying  leaves  half  of  his  property  to  his  wife, 
one-sixth  to  each  of  two  daughters,  one-twelfth  to  a  servant, 
and  the  remaining  $000  to  the  poor :  what  was  the  amount 
)f  his  property  1 


iOS  ELEMENTARY     ALGEBRA. 

Denote  the  amount  of  the  property  by  x. 

x 
Then         —  =         -what  he  left  to  his  wife, 


what  he  left  to  one  daughter, 


and  — -  =  —     what  he  left  to  both  daughters , 

D  O 

X 

also  —  =         what  he  left  to  his  servant. 

and  $600  what  he  left  to  the  poor. 

Then,  by  the  conditions 

—  +  —  +  —  +  600  =  x  the  amount  of  the  property, 

which  gives     x  =  $7200. 

10.  A  and  B  play  together  at  cards.  A  sits  down  with 
$84  and  B  with  $48.  Each  loses  and  wins  in  turn,  when 
it  appears  that  A  has  five  times  as  much  as  B.  How  mujh 
did  A  win  1 

Let  x  represent  what  A  won. 
Then,  A     rose  with     $84  +  x     dollars, 

and  B     rose  with     $48  —  x     dollars. 

But  by  the  conditions,  we  have 

84  +  x  =  5(48  —  x), 
hence,  84  +  x  =  240  —  5x ; 

and,  Gx  =  156, 

consequently,  x  —  $26     what  A  won. 

Verification. 

84-}-26  =  110;     48-26  =  22; 
110  =  5(22)  =  110. 


EQUATIONS     OF     THE     FIRST     DEGREE.        109 

11.  A  can  do  a  piece  of  work  alone  in  10  days,  B  in  13 
days ;  in  what  time  can  they  do  it,  if  they  work  together? 

Denote  the  time  by  x,  and  the  work  by  1.     Then,  in 

1  day,  A  can  do  —  of  the  work,  and 

B  can  do  —  of  the  work ;  and  in 

lo 

X 

x  days,  A  can  do  — -  of  the  work,  and 
J  '  10  ' 

X 

B  can  do  — ;  of  the  work ; 
lo 

hence,  by  the  conditions 

~  +  ^-  =  1,     which  gives     13x  -f  lOz  =  130  : 

1U         lo 

130 
hence,         23a;  =  130,    x  —  — -  =  5^f  days. 

12.  A  fox,  pursued  by  a  greyhound,  has  a  start  of  CO 
leaps.  He  makes  9  leaps  while  the  greyhound  makes  but 
6 ;  but,  3  leaps  of  the  greyhound  are  equivalent  to  7  of  the 
fox.  How  many  leaps  must  the  greyhound  make  to  over- 
take the  fox  1 

From  the  enunciation,  it  is  evident  that  the  distance  to 
be  passed  over  by  the  greyhound  is  composed  of  the  60 
leaps  which  the  fox  is  in  advance,  plus  the  distance  that  the 
fox  passes  over  from  the  moment  when  the  greyhound  starts 
in  pursuit  of  him.  Hence,  if  we  can  find  the  expressions 
for  these  two  distances,  it  will  be  easy  to  state  the  problem. 

Let  x  =  the  number  of  leaps  made  by  the  greyhound 
before  he  overtakes  the  fox. 

Now,  since  the  fox  makes  9  leaps  while  the  greyhound 

9  3 

makes  but  G,  tV  fox  will  make    —   or    —   leaps   while 


110  ELEMENTARY     ALGKBKA. 

the  greyhound  makes  1  ;  and,  therefore,  while  the  gre^houn,' 

ox 
makes  x  leaps,  the  fox  will  make    —     leaps. 

Hence,  the  distance  which  the  greyhound  must  pass  ovej' 

Sx 
will  be  expressed  by  GO  +  —    leaps  of  the  fox. 

It  might  be  supposed,  that  the  equation  might  be  obtained 

by  merely  placing  x  equal  to  60  +  —  x  ;    but  in  doing  so,  a 

manifest  error  would  be  committed  ;  for  the  leaps  of  the  grey- 
hound are  greater  than  those  of  the  fox,  and  we  should  then 
equate  numbers  of  different  denominations ;  that  is,  num- 
bers having  different  units.  Hence,  it  is  necessary  to  ex- 
press the  leaps  of  the  fox  in  terms  of  those  of  the  grey- 
hound, or  reciprocally.  Now,  according  to  the  enunciation 
3  leaps  of  the  greyhound  are  equivalent  to  7  leaps  of  the 

7 
fox  ;  then,  1  leap  of  the  greyhound  is  equivalent  to  —  leaps 

o 

of  the  fox  ;  and  consequently,  x  leaps  of  the  greyhound  are 

7,r 
equivalent  to  — -  of  the  fox's  leaps. 
o 

Hence,  we  have  the  equation 

T  =  60  +  -*. 

Making  the  denominators  disappear 

14.c  =  300  +  9a:, 
whence,  5x  —  300     and     x  ■=.  72  : 

Therefore,  the  greyhound  will  make  72  leaps  before  over 
taking  the  fox,  and  during  this  time,  the  fox  will  make 

3 

72  X  —    or    10S  leaps. 


EQUATIOK8     O  V     T  H  K     V I  K  !i  T     DKG11KE.         ID 

Verification. 

The  72  leaps  of  the  greyhound  are  equivalent  to 

72  X  7 

— - —  =  1G8  leaps  of  the  fox, 

<o 

and  GO  +  10S  =  1G8, 

the  leaps  which  the  fox  made  from  the  beginning. 

13.  A  father  leaves  his  property,  amounting  to  §2520,  to 
four  sons,  A,  B,  C,  and  D.  C  is  to  have  §300,  B  as  much 
as  C  and  D  together,  and  A  twice  as  much  as  B  less  $1000" 
how  much  do  A,  B,  and  D  receive  ? 

Ans.  A,  $700,  B,  §SS0,  D,  §520 

14.  An  estate  of  $7500  is  to  be  divided  between  a  widow 
two  sons,  and  three  daughters,  so  that  each  son  shall  receive 
twice  as  much  as  each  daughter,  and  the  widow  herself  §500 
more  than  all  the  children  :  what  was  her  share,  and  what 
the  share  of  each  child  1 

(  Widow's  share,    $4000. 

Ans.  }  Each  son's,  §1000. 

(  Each  daughter's,  $  500. 

15.  A  company  of  180  persons  consists  of  men,  women, 
and  children.  The  men  are  8  more  in  number  than  the 
women,  and  the  children  20  more  than  the  men  and  women 
together:  how  many  o<*  each  sort  in  the  company  % 

Ans.  44  men,  3G  women,  100  children. 

10.  A  father  divides  §2000  among  five  sons,  so  that  each 
elder  should  receive  §40  more  than  his  next  younger  bro- 
ther :  what  is  the  share  of  the  youngest  ?  Ans.  §320. 

17.  A  purse  of  §2850  is  to  be  divided  among  three  per- 
sons, A,  B,  and  O.     A's  share  is  to  bo  to  B's  aa  0  to  11 


LiS  ELEMENTARY     ALGEBRA. 

and  C  is  to  have  $300  more  than  A  and  B  together :   what 
is  each  one's  share  1      Ans.  A's  $450,  B's  $825,  C's  $1575. 

18.  Two  pedestrians  start  from  the  same  point ;  the  first 
steps  twice  as  far  as  the  second,  but  the  second  makes  5 
steps  while  the  first  makes  but  one.  At  the  end  of  a  cer- 
tain time  they  are  300  feet  apart.  Now,  allowing  each  of 
the  longer  paces  to  be  3  feet,  how  far  will  each  have  trav- 
elled ?  Ans.  1st,  200  feet;  2d,  500. 

19.  Two  carpenters,  24  journeymen,  and  8  apprentices, 
received  at  the  end  of  a  certain  time  $144.  The  carpen- 
ters received  $1  per  day,  each  journeyman,  half  a  dollar, 
and  each  apprentice  25  cents  :  how  many  days  were  they 
employed  1  Ans.  9  days. 

20.  A  capitalist  receives  a  yearly  income  of  $2940  :  four- 
fifths  of  his  money  bears  an  interest  of  4  per  cent,  and  the 
remainder  of  5  per  cent :  how  much  has  he  at  interest  1 

Ans.  70000. 

21.  A  cistern  containing  60  gallons  of  water  has  three 
unequal  cocks  for  discharging  it ;  the  largest  will  empty  it 
in  one  hour,  the  second  in  two  hours,  and  the  third  in  three : 
hi  what  time  will  the  cistern  be  emptied  if  they  all  run 
together  1  Ans.  32^  min. 

22.  In  a  certain  orchard,  one-half  are  apple  trees,  one- 
fourth  peach  trees,  one-sixth  plum  trees  ;  there  are  also,  120 
cherry  trees,  and  80  pear  trees :  how  many  trees  in  the 
orchard  ?  Am.  2400. 

23.  A  farmer  being  asked  how  many  sheep  he  had,  an- 
swered, that  he  had  them  in  five  fields  ;  in  the  1st  he  had  J, 
in  the  2d,  ^,  in  the  3d,  -|-,  and  in  the  4th,  ^,  and  in  the  5th, 
450  :   how  many  had  he  1  Ans.  1200. 

24.  My  horse  and  saddle  together  are  worth  $132,  and 
the  horse  is  worth  ten  times  as  much  as  the  saddle  :  what 
is  tho  value  of  the  horse1?  Ans.  120. 


EQUATIONS     OF     THE     FIRST     DEUKKE.         113 

25.  The  rent  of  an  estate  is  this  year  8  per  cent  greater 
than  it  was  last.  This  year  it  is  $1890  :  what  was  it  last 
year  %  Ans.  $1750. 

26.  What  number  is  that  from  which,  if  5  be  subtracted, 
|  of  the  remainder  will  be  40  %  Ans.  65. 

27.  A  post  is  |-  in  the  mud,  ^  in  the  water,  and  10  feet 
ibove  the  water :  what  is  the  whole  length  of  the  post  1 

Ans.  24  feet. 

28.  After  paying  \  and~|  of  my  money,  1  had  GO  guineas 
left  in  my  purse  :  how  many  guineas  were  in  it  at  first  1 

Ans.  120. 

29.  A  person  was  desirous  of  giving  3  pence  apiece  to 
some  beggars,  but  found  he  had  not  money  enough  in  his 
pocket  by  8  pence ;  he  therefore  gave  them  each  2  pence 
and  had  3  pence  l-emaining  :  required  the  number  of  beg- 
gars. Ans.  11. 

30.  A  person,  in  play,  lost  \  of  his  money,  and  then  won 
3  shillings ;  after  which  he  lost  ^  of  what  he  then  had  ;  and 
this  done,  found  that  he  had  but  12  shillings  remaining: 
what  had  he  at  first  ?  Ans.  20s. 

31.  Two  persons,  A  and  B,  lay  out  equal  sums  of  money 
in  trade  ;  A  gains  $120,  and  B  loses  $87,  and  A's  money 
is  now  double  of  B's  :  what  did  each  lay  out  ?     Ans.  $300. 

32.  A  person  goes   to   a   tavern   with  a  certain  sum  of 

money  in  his  pocket,  where  he  spends  2  shillings  :  he  then 

borrows  as  much  money  as  he  had  left,  and  going  to  another 

tavern,  he  there  spends  2  shillings  also ;    then  borrowing 

again  as  much  money  as  was  left,  he  went  to  a  third  tavern, 

where  likewise  he  spent  2  shillings  and  borrowed  as  much 

as  he  had  left ;  and  again  spending  2  shillings  at  a  fourth 

tavern,  he  then  had  nothing  remaining.     What  had  he  at 

first  ?  .  Am.  3s.  9d. 

6 


114  ELEMENTARY     ALUEBKA. 

Of  Equations  of  the  First  Degree  involving  two  o>  mort 
unknown  quantities. 

72.  Several  of  the  problems  already  discussed  have 
apparently  involved  more  than  one  unknown  quantity  ;  yet 
we  have  been  able  to  solve  them  all  by  the  aid  of  a  single 
unknown  symbol.  In  these  cases,  the  required  parts  of  the 
problem  have  been  so  connected  that  we  have  been  able  to 
express  the  relations  between  them  by  means  of  a  single 
equation.  We  come  now  to  those  problems,  in  the  solution 
of  which,  we  employ  more  than  one  unknown  quantity. 

Let  us  first  examine  some  of  those  problems  which  we 
have  already  solved  by  the  aid  of  but  a  single  unknown 
symbol. 

1.  Given  the  sum  of  two  numbers  equal  to  36,  and  their 
difference  equal  to  12,  to  find  the  numbers. 

Let  x  =  the  greater,  and  y  =  the  less  number. 
Then,  from  the  1st  condition     .     .     .     .     x  -f-  y  =  36, 
and  from  the  second,      .     .     .     .     .     .     .     x  —  y  —  12. 

By  adding  (Art.  65,  Ax.  1),     ....         2  x  =  48. 
By  subtracting  (Art.  65,  Ax.  2),  .     .     .  2  y  —  24. 

Each  of  these  equations  contains  but  one  unknown  quan- 
tity. 

48 
From  the  first,  we  obtain x  =  —  —  24. 

24 

And  from  the  second,      .     .     .     .     .     .  y  =  —  =  12. 

Verification. 

x  -}-  y  =  36     gives     24  4-  12  =  36, 
x-y-YZ        "        24 -12  =.12. 


EQUATIONS     OF     THE     FIRST     DEGREE.         115 

General  Solution. 
Let  x  =  the  greater,  and  y  the  less  number. 

Then  "by  the  conditions x  ~f-  y  =  a, 

and x  —  y  =  b. 

By  adding,  (Art.  65,  Ax.  1), 2x  =  a  +  b. 

By  subtracting,  (Art.  65,  Ax.  2),   .     .     .     .  2y  =  a  —  b. 
Each  of  these  equations  contains  but  one  unknown  quantity. 

_           ,                        ,  a  +  b 

rrom  the  first,  we  obtain x  =  — - — • 

And  from  the  second, y  =  — - — . 

'  *  2 

a-\-  b       a  —  b        2a                 ,    a  -{-  6        O  —  S       26       , 
-2-+-2-  =  y^a;and   _ —  =  -  =  5. 

For  a  second  example,  let  us  also  take  a  problem  that 
has  been  already  solved. 

2.  A  person  engaged  a  workman  for  48  days.  For  each 
day  that  he  labored  he  was  to  receive  24  cents,  and  for  each 
day  that  he  was  idle  he  was  to  pay  12  cents  for  his  board. 
At  the  end  of  the  48  days  the  account  was  settled,  when  the 
laborer  received  504  cents.  Required  the  number  of  work 
ing  days,  and  the  number  of  days  he  was  idle. 
Let  x  =     the  number  of  working  days, 

y  =     the  number  of  idle  days. 
Then  24a;  =     what  he  earned, 

and  \2y  =     what  he  paid  for  his  board. 

Then,  by  the  conditions  of  the  question,  we  have 

•  x  +  y      =48, 
and  24x—  12y  =  504. 

This  is  the  statement  of  the  pr<dl»lem. 


116  ELEMENTARY     ALGE11RA. 

It  has  already  been  shown  (Art.  65,  Ax.  3),  that  the  two 
members  of  an  equation  can  be  multiplied  by  the  same 
number,  without  destroying  the  equality.  Let,  then,  the 
first  equation  be  multiplied  by  24,  the  co-efficient  of  x  in 
the  second :  we  shall  then  have 

24r  +  24y  =  1152, 
24z  —  12y  =    504, 

And  by  subtracting,  86?/  =    648, 

A  648  1H 

and  y  =  —  =  18. 

Substituting  this  value  of  y  in  the  equation 

24z  —  12y  =  504,     we  have     24.r  —  216  =  504, 
which  gives 

24<r  =  504  +  216  =  720,     and     x  =  —^  =  30. 

Verification. 

•x  +      y  =    48     gives  30  4-  18  =    48, 

24*  —  12y  =  504     gives     24  X  30  —  12  X  18  =  504. 

Elimination. 

73.  The  process  of  combining  two  or  more  equations,  in- 
volving two  or  more  unknown  quantities,  and  deducing  there- 
from a  single  equation  involving  but  one,  is  called  elimina- 
tion. 

73.  "What  is  elimination  ?  How  many  methods  of  elimination  are 
there  ?  Give  the  rule  for  elimination  by  addition  and  subtraction.  Wlmt 
is  the  fust  alep  ?     What  the  nreorid  ?     "What  tho  third  f 


EQUATIONS     OF     THE     FIHST     DEGSKE  117 

There  are  three  principal  methods  of  elimination : 

1st.  By  addition  and  subtraction. 

2d.    By  substitution. 

3d.    By  comparison. 
We  will  consider  these  methods  separately. 

Elimination  by  Addition  and  Subtraction. 

1.  Take  the  twn  equations 

Sx  -  2y  =  7 
8x  +  2y  =  48. 

If  we  add  these  two  equations,  member  to  member,  we 
obtain 

11a;  =  55: 

which  gives  by  dividing  by  11 

x  =  5  : 

and  substituting  this  value  in  either  of  the  given  equations, 
we  find 

V  =4. 

2.  Again,  take  the  equations 

8x  +  2y  =  48 
Sx  +  2y  =  23. 

If  we  subtract  the  2d  equation  from  the  first,  we  obtain 

5x  =  25, 

which  gives,  by  dividing  by  5, 

x  =  5 : 
and  by  substituting  this  value,  we  find 
y  =  4. 


118  ELEMENTARY     ALGEBKA, 

8.   Take  the  two  equations 

5x  +  7y  =  43. 
llar  +  9y  =  G9. 

If,  in  these  equations,  one  of  the  unknown  quantities  was 
affected  with  the  same  co-efficient,  we  might,  by  a  simple 
subtraction,  form  a  new  equation  which  would  contain  but 
one  unknown  quantity. 

Now,  if  both  members  of  the  first  equation  be  multiplied 
by  9,  the  co-efficient  of  y  in  the  second,  and  the  two  mem- 
bers of  the  second  by  7,  the  co-efficient  of  y  in  the  first,  we 
will  obtain 

45x  +  63y  =  387, 
77z  +  63y  =  483. 

Subtracting,  then,  the  first  of  these  equations  from  the 
second,  there  results 

o'2x  =  96,     whence     x  =  o. 

Again,  if  we  multiply  both  members  of  the  first  equation 
by  11,  the  co-efficient  of  a;  in  the  second,  and  both  members 
of  the  second  by  5,  the  co-efficient  of  x  in  the  first,  we  shall 
form  the  two  equations 

55*  +  THy  =  473, 
55.c  +  45y  =  345. 

Subtracting,  then,  the  second  of  these  two  equations  from 
the  first,  there  results 

32y  =  128,     whence     y  =  4. 
Therefore  x  =.  3  and  y  =  4,  are  the  values  of  x  and  y. 

Verification. 

5x  ■+  ly  =  43    gives      5x3  +  7x4=  15 +  28  =  43; 
11*4-05  =  69      "        11x3  +  9x4  =  33  +  36  =  69. 


EQUATIONS     OF     THE     FIRST     DEGREE.         119 

The  method  of  elimination  just  explained,  is  called  the 
method  by  addition  and  subtraction. 

To  eliminate  by  this  method  we  have  the  following 

RULE. 

I.  See  which  oj  the  unknown  quantities  you  will  eliminate. 

II.  Make  the  co-efficients  of  this  unknown  quantity  equal  in 
the  two  equations,  either  by  multiplication  or  division. 

III.  If  the  signs  of  the  like  terms  are  the  same  in  both 
equations,  subtract  one  equation  from  the  other  ;  but  if  the 
signs  are  unlike,  add  them. 

EXAMPLES. 

4.  Find  the  values  of  x  and  y  in  the  equations 

Zx  —  y  =  3, 
y  +  2x  =  7. 

Ans.  x  =  2,  y  =  3. 

5.  Fiiid  the  values  of  x  and  y  in  the  equations 

Ax  -  7y  =  -  22, 
5x  +  2y=z  37. 

Ans.  x  =  5,  y  =  6. 

6.  Find  the  values  of  x  and  y  in  the  equations 

2x  -f  6y  =  42, 
8x  —  6y=    3. 

Ans.  x  =zA\,  y  —  5|. 

7.  Find  the  values  of  x  and  y  in  the  equations 

&c  — 9y  =  1, 
&c  —  3y  =  4#. 

,4ns.  x  =  ^,  y  =  £. 


120  ELEMENTARY     ALGEBUA. 

8.  Find  the  values  of  x  and  y  in  the  equations 

14z  -  15y  =  12, 
7x  A-    8y  =  37. 

^l»s.  x  =  3,  y  =•  9 

9.  Find  the  values  of  a;  and  y  in  the  equations 

1  1 


-x  +  -y  =  Q>\. 


1       .     X 
—  x  A 

3^2 

Ans.  x  =  6,  y  =  9. 

10.  Find  the  values  of  x  and  y  in  the  equations 

1  1 

x  —  y  =i  —  2. 

.4ns.  as  =  14,  y  =  16. 

11.  Says  A  to  B,  you  give  me  $40  of  your  money,  and 
I  shall  then  have  5  times  as  much  as  you  will  have  left. 
Now  they  both  had  $120  :  how  much  had  each? 

Ans.  Each  had  $60. 

12.  A  father  says  to  his  son,  "  twenty  years  ago,  my  age 
was  four  times  yours ;  now  it  is  just  double  :"  what  were 
their  ages  1  .       j  Father's,  60  years. 


Son's,  30  years. 
13.  A  father  divides  his  property  between  his  two  sons. 
At  the  end  of  the  first  year  the  elder  had  spent  one-quarter 
of  his,  and  the  younger  had  made  $1000,  and  their  property 
was  then  equal.  After  this  the  elder  spends  $500  and  the 
younger  makes  $2000,  when  it  appears  the  younger  has  just 
double  the  elder  :  what  had  each  from  the  father  1 


,      j  Elder,       $4000. 
AnSm  \  Younger,  $2000. 


EQUATIONS     OF     THE     FIRST     DEGREE.         121 

14.  If  John  give  Charles  15  apples,  they  will  hav«.  the 
same  number;  but  if  Charles  give  15  to  John,  John  will 
have  15  times  as  many,  wanting  10,  as  Charles  will  have 
left.     How  many  had  each  ?  .        j  John       50. 

(  Charles  20. 

15.  Two  clerks,  A  and  B,  have  salaries  which  are  together 
equal  to  $900.  A  spends  yL  per  year  of  what  he  receives, 
and  B  adds  as  much  to  his  as  A  spends.  At  the  end  of  the 
year  they  have  equal  sums  :  what  was  the  salary  of  each  ? 

Ans.   \  A's  =  50°- 
400 


j  A's  = 
(B's  = 


Elimination  by  Substitution. 

74.  Let  us  again  take  the  equations 

5* +7*,  =  43, 
llx4-9y  =  69. 

Find  the  value  of  x  in  the  first  equation,  which  gives 

43-7y 
X~        5 

Substitute  this  value  of  x  in  the  second  equation,  and  we 
have 

nx^  +  o^eg, 

or,  473  -  77y  +  45y  =  345, 

or,  —  32y  =  —  128. 

Hence,  y  =  4, 

,  43-28      „ 

and,  x  — —  =  3. 

5 

0 


122  ELEMENTARY     ALGEBRA. 

This  method  is  called  the  method  by  substitution :  we 
have  fer  the  process  the  following 

RULE. 

Find  the  value  of  one  of  the  unknown  quantities  from 
either  of  the  equations  ;  then  substitute  this  value  in  the  other 
equation:  there  will  thus  arise  a  new  equation  with  but  one 
unknown  quantity. 

Remark. — This  method  of  elimination  is  used  to  great 
advantage  when  the  co-efficient  of  either  of  the  unknown 
quantities  is  unity.  £ 

EXAMPLES. 

1.  Find,  by  the  last  method,  the  values  of  x  and  y  in  the 
equations 

Bx  —  y  =  1     and     By  —  2x  =  4. 

Ans.  x  =  1,  y  =  2. 

2.  Find  the  values  of  x  and  y  in  the  equations 

by  —  Ax  =  —  22     and     By  +  4x  =  38. 

Ans.  x  =  8,  y  =  2. 

\  Find  the  values  of  x  and  y  in  the  equations 
x  +  Sy  =  18     and     y  —  Bx  =  —  29. 

Ans.  x  =  10,  y  =  1. 

i.  Find  the  values  of  x  and  y  in  the  equations 

2 

5x  —  y  =  13     and    8z  4-  —  y  =  29. 

Ans.  x  =  3^,  y  =  4j. 

7      Give  the  rule  for  elimination  by  substitution  ?     When  is  it  desira- 
ble to  use  this,  method  ! 


EQUATIONS    OF     THE     FIBUT     DBUKKK.         123 

5.  Find  the  values  of  x  and  y,  from  the  equations 

l(kr  —  -|-  =  G9     and     lOy  —  ~=  49. 
5  s       7 

Ans.  x  =  7,  y  =  5. 

6.  Find  the  values  of  x  and  y,  from  the  equations 

*+o-*  —  T  =  10     and     i  +  TK  =  2- 

2  5  8        10 

^4n5.  a;  =  8,  y  =  10. 
1.  Find  the  values  of  x  and  y,  from  the  equations 

f.-|+5=S,     *  +  |  =  17f 

-4ns.  a;  =  15,  y  =  14. 


8.  Find  the  values  of  x  and  y,  from  the  equations 

-+3+3  =  61     and     T-T=-. 

-4ns.  a;  =  3f ,  y  =  4. 

9.  Find  the  values  of  x  and  y,  from  the  equations 

f--  +  6  =  5,     and    g-i  =  «. 

Ans.  x  —  12,  y  =  10. 

10.  Find  the  values  of  x  and  y,  from  the  equations 

f-y-l  =  -9,     and     5*-g  =  29. 

,4ns.  a;  =  6,  y  =  7. 

11.  Two  misers,  A  and  B,  sit  down  to  count  over  their 
money.  They  both  have  $20000,  and  B  has  three  times  as 
much  as  A  :  how  much  has  each  1 

A . .     $5000. 
B  .  .   $15000. 


124  ELEMENTARY     ALGEBRA. 

12.  A  person  has  two  purses.  If  he  puts  $7  into  the  first 
purse,  it  is  worth  three  times  as  much  as  the  second :  but  if 
he  puts  $7  into  the  second,  it  becomes  worth  five  times  aa 
much  as  the  first :  what  is  the  value  of  each  purse  1 

Ans.  1st,  $2  :  2d,  $3. 

13.  Two  numbers  have  the  following  properties:  if  the 
first  be  multiplied  by  6,  the  product  will  be  equal  co  the 
second  multiplied  by  5 ;  and  one  subtracted  from  the  first 
leaves  the  same  remainder  as  2  subtracted  from  the  second  : 
what  are  the  numbers  1  Ans.  5  and  6. 

14.  Find  two  numbers  with  the  following  properties :  the 
first  increased  by  2  to  be  3|  times  as  great  as  the  second : 
and  the  second  increased  by  4  gives  a  number  equal  to  half 
the  first :  what  are  the  numbers  1  Ans.  24  and  8. 

15.  A  father  says  to  his  son,  "twelve  years  ago,  I  was 
twice  as  old  as  you  are  now :  four  times  your  age  at  that 
time,  plus  twelve  years,  will  express  my  age  twelve  years 
hence :"  what  were  their  ages  1        .         j  Father,  72  years. 

"*  Son,        30      " 

Elimination  by  Comparison. 

75.  Take  the  same  equations 

5x  +  7y  =  43, 
lla:4-  9y  =  69. 

Finding  the  value  of  x  from  the  first  equation,  we  have 
43  —  ~iy 
*  =  — 5— 
and  finding  the  value  of  x  from  the  second,  we  obtain 
69  —  9y 


EQUATIONS     OF     THE     FIRST     DEGREE.         125 

Let  these  two  values  of  x  be  placed  equal  to  each  other, 
and  we  have 

43  -  ly  _  69  -  9y 

5        "~       11 

Or,  473  -  77y  =  345  -  45y  ; 

or,  —  32y  =  —  128. 

Hence,  y  =  4. 

G9-3G      „ 
And,  x  =  — -- —  =  3. 

This  method  of  elimination  is  called  the  method  by  com 
parison,  for  which  we  have  the  following 

EULE. 

I.  Find  the  value  of  the  same  unknown  quantity  from  each 
equation. 

II.  Place  these  values  equal  to  each  other ;  and  a  new  equa- 
tion will  arise  involving  but  one  unknown  quantity. 

EXAMPLES. 

1.  Find,  by  the  last  rule,  the  values  of  x  and  y,  from  the 
equations 

3*  +  -r  +  6=42     and     y_^  =  14£ 


22 


nr* 


Am.  x  =.  11,  y  =  15. 


75^  Give  the  rule  for  elimination  by  comparison!     What  is  the  Ural 
rtep       What  the  eecond ! 


126  ELEMENTARY     ALGEBRA. 

2.  Find  the  values  of  x  and  y,  from  the  equations 

^.-^4-5  =  6    and    f-  +  4  =  ^  +  6. 

4        7  5  14 

Ans.  x  ==  28,  y  =  20. 

8.  Find  the  values  of  x  and  y,  from  the  equations 

V        x    ,   22       ,  .,     _  rt 

tz r  +  -r-  =  !     and     3y  —  a;  =  6. 

10       4        8  y 

^Ins.  a:  =  9,  y  =  5. 

4.  Find  the  values  of  x  and  y,  from  the  equations 

y-3=-a4-5     and     _T*  =  y  _  3J. 

^4ns.  x  =  2,  y  =  9. 

5.  Find  the  values  of  x  and  y,  from  the  equations 

V  —  XX  XV 

-4?i5.  a;  =  16,  y  =  7. 

6.  Find  the  values  of  x  and  y,  from  the  equations 

y  -\-  x      y  —  x  2y  ., 

— t; 1 5 —  =  *;  —  -<->    and    a;  +  y  =  16. 

^4ns.  x  —  10,  y  =  6, 

7.  Find  the  values  of  a:  and  y,  from  the  equations 

2x  —  3y  ».  y  —  1 

-4ns.  a;  =  1,  y  =  3 


8.  Find  the  values  of  a;  and  y,  from  the  equations 

-4  x 

^ns,  a;  =  10,  y  .»  13. 


x    •  4  ic 

2j,  +  3a;  =  y  +  43,    y —  =  y  —  — . 


EQUATIONS     OF     THE     FIRST     DEGREE.         127 

9.  Find  the  values  of  x  and  y,  from  the  equations 

4y — -^-  =  x  +  18,  and  27  —  y  =  a;  -f  y  -f-  4. 

Ans.  x  =z  9,  y  =  7. 
10.  Find  the  values  of  x  and  y,  from  the  equations- 

^4»5.  a?  =  10,  y  =  20. 

76.  Having  explained  the  principal  methods  of  elimina- 
tion, we  shall  add  a  few  examples  which  may  be  solved  by 
any  one  of  them  ;  and  often  indeed,  it  may  be  advantageous 
to  employ  them  all  even  in  the  same  problem. 

GENERAL    EXAMPLES. 

1.  Given  2x  +  3y  =  16,  and  Sx  —  2y  =  11,  to  find  the 
values  of  x  and  y.  Ans.  x  =  5,  y  =  2. 

_    ~.         2x      3y        9        .    3x      2y        01 

2-GlvenT  +  T=20and  T  +  T  =  l20'    t0  fiBd 

1  1 

the  values  of  x  and  y.  Ans.  x  =  —  5    y  =  — • 

y  2*3 

3.  Given  j  +  7y  =  99,  and  |-  +  7*  =  51,  to  find  the 
values  of  x  and  y.  Ans.  x  =  7,  y  =  14. 

4.  Given 

U   rmd  the  values  of  x  and  y.  ./ins.  a;  =  60,  y  =•  40. 


128  ELEMENTARY     ALGEBRA. 

QUESTIONS. 

1.  What  fraction  is  that,  to  the  numerator  of  which  if  J 

be  added,  the  value  will  be  — ,     but  if  one  be  added  to  ite 

o 

1   , 

denominator,  the  value  will  be  — -  • 

4 

x 

Let  the  fraction  be  represented  by    — • 

Then  by  the  conditions 

x+l  \  .  x  1 

=  — ,     and,     — — -  =  —  . 

V  3  y  +  1        4 

Whence,  ox  -j-  3  =  y,     and,     4x  =  y  +  1. 

Therefore,  by  subtracting, 

x  —  3  =  1     and     x  =  4. 

Hence,  12  +  3  =  ?/ ; 

therefore,  y  =  15. 

2.  A  market-woman  bought  a  certain  number  of  eggs  at 
2  for  a  penny,  and  as  many  others,  at  3  for  a  penny ;  and 
having  sold  them  altogether,  at  the  rate  of  5  for  2d,  found 
that  she  had  lost  4c? :  how  many  of  both  kinds  did  she  buy  1 

Let  2x  =     the  whole  number  of  eags. 

Then  x  =     the  number  of  eggs  of  each  sort. 

1 

Then  will  —  x  =     the  cost  of  the  first  sort, 

and  —  x  =     the  cost  of  the  secohd  sort. 

o 


But  by  the  conditions  of  the  question     5  :  2x  :  :  2 
the  amount  for  which  the  eggs  were  sold. 


4x 
5" 


EQUATIONS     OF     THE     FIRST     DEGREE.         120 

Hence,  by  the  conditions 

11  4x 

Therefore,  15*  +  10*  —  24*  =  120 

or  x  —  120  ; 

the  number  of  eggs  of  each  sort. 

3.  A  person  possessed  a  capital  of  30,000  dollars,  for 
which  he  drew  a  certain  interest ;  but  he  owed  the  sum  of 
20,000  dollars,  for  which  he  paid  a  certain  interest.  The 
interest  that  he  received  exceeded  that  which  he  paid  by 
800  dollars.  Another  person  possessed  35,000  dollars,  for 
which  he  received  interest  at  the  second  of  the  above  rates ; 
but  he  owed  24,000  dollars,  for  which  he  paid  interest  at 
the  first  of  the  above  rates.  The  interest  that  he  received 
exceeded  that  which  he  paid  by  310  dollars.  Required  the 
two  rates  of  interest. 

Let  x  and  y  denote  the  two  rates  of  interest ;  that  is,  the 
interest  of  $100  for  the  given  time. 

To  obtain  the  interest  of  $30,000  at  the  first  rate,  denoted 
by  x,  we  form  the  proportion 

100  :  x  :  :  30,000    :    B-^^-     or     300*. 
100 

And  for  the  interest  $20,000,  the  rate  being  y, 

100  :  y  :  :  20,000    :    2Q'^?°y     or     200y. 

But  by  the  conditions,  the  difference  between  these  two 
amounts  is  equal  to  800  dollars. 

We  hav?,  then,  for  the  first  equation  of  the  problem 

300*  —  200y  =  800. 
(}* 


130  ELEMENTARY     ALGEBRA. 

By  expressing  algebraically,  the  second  condition  of  the 
problem,  we  obtain  the  other  equation, 

350y  —  240a;  =  310. 

Both  members  of  the  first  equation  being  divisible  by 
100,  and  those  of  the  second  by  10,  we  have 

3*  —  2y  =  8,         35y  -  2±x  =  31. 

To  eliminate  x,  multiply  the  first  equation  by  8,  and  then 
add  the  result  to  the  second  :  there  results 

19y  =  95,     whence     y  =  5. 

Substituting  for  y,  in  the  first  equation,  this  value,  and 
that  equation  becomes 

3a;  —  10  =  8,     whence     x  =  6. 

Therefore,  the  first  rate  is  G  per  cent,  and  the  second  5. 

Verification. 

$30,000,    placed  at  G  per  cent,  gives     300  x  G  =  Si 800. 
$20,000,        "  5         "  "       200  x  5  =  $1000. 

And  we  have  1800  —  1000  =  800. 

The  second  condition  can  be  verified  in  the  same  manner. 

4.  What  two  numbers  are  those,  whose  difference  is  7, 
and  sum  33  1  Am.  13  and  20. 

5.  To  divide  the  number  75  into  two  such  parts,  that 
three  times  the  greater  may  exceed  seven  times  the  less  by 
15.      t  *Ans.  54  and  21. 

G.  In  a  mixture  of  wine  and  cider,  \  of  the  whole  plus 
25  gallons  was  wine,  and  \  part  minus  5  gallons  was  cider : 
h'  »w  many  gallons  were  there  of  each  ? 

Ans.  85  of  wine,  and  35  of  cider. 


EQUATIONS     OF     THE     FIRST     DEGREE.         131 

7.  A  bill  of  £120  was  paid  in  guineas  and  moidores,  and 
the  number  of  pieces  used,  of  both  sorts,  was  just  100.  If 
the  guinea  be  estimated  at  21s,  and  the  moidore  at  27s,  how 
many  pieces  were  there  of  each  sort  ?  Ans.  50  of  each. 

8.  Two  travellers  set  out  at  the  same  time  from  London 
and  York,  whose  distance  apart  is  150  miles.  One  of  them 
goes  8  miles  a  day,  and  the  other  7  :  in  what  time  will  they 
meet?  Ans.  In  10  days. 

9.  At  a  certain  election,  375  persons  voted  for  two  candi- 
dates, and  the  candidate  chosen  had  a  majority  of  91 :  how 
many  voted  for  each  ? 

Ans.  233  for  one,  and  142  for  the  other. 

10.  A  person  has  two  horses,  and  a  saddle  worth  £50. 
Now,  if  the  saddle  be  put  on  the -back  of  the  first  horse,  it 
will  make  his  value  double  that  of  the  second ;  but  if  it  be 
put  on  the  back  of  the  second,  it  will  make  his  value  triple 
that  of  the  first.     What  is  the  value  of  each  horse'? 

Ans.  One  £30,  and  the  other  £40. 

11.  The  hour  and  minute  hands  of  a  clock  are  exactly 
together  at  12  o'clock:  when  will  they  be  again  together? 

Ans.  Ik.  5^rm. 

12.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer 
in  12  days ;  but  when  the  man  was  from  home,  it  lasted 
the  woman  30  days  :  how  many  days  would  the  man  alone 
be  in  drinking  it  ?  Ans.  20  days. 

13.  If  32  pounds  of  sea-water  contain  1  pound  of  salt, 
how  much  fresh  water  must  be  added  to  these  32  pounds, 
in  order  that  the  quantity  of  salt  contained  in  32  pounds  of 
the  new  mixture  shall  be  reduced  to  2  ounces,  or  ^  of  a 
pound  ?  Ans.  224/6. 

14.  A  person  who  possessed  100,000  dollars,  placed  the 
greater  part  of  it  out  at  5  per  cent  interest,  and  the  other 


132  ELEMENTARY     ALGEBRA. 

at  4  per  cent.     The  interest  which  he  received  for  the  whole 
amounted  to  4840  dollars.     Required  the  two  parts, 

Ans.  64,000  and  30,000. 
15.  At  the  close  of  an  election,  the  successful  candidate 
had  a  majority  of  1500  votes.  Had  a  fourth  of  the  votes 
of  the  unsuccessful  candidate  been  also  given  to  him,  he 
would  have  received  three  times  as  many  as  his  competitor, 
wanting  three  thousand  five  hundred  :  how  many  votes  did 
each  receive  1  ,       j  1st,  6500. 

^nS,(2d,  5000. 
15.  A  gentleman  bought  a  gold  and  a  silver  watch,  and 
a  chain  worth  $25.  When  he  put  the  chain  on  the  gold 
watch,  it  and  the  chain  became  worth  three  and  a  half  times 
more  than  the  silver  watch ;  but  when  he  put  the  chain  on 
the  silver  watch,  they  became  worth  one-half  the  gold  watch 
and  15  dollars  over :  what  was  the  value  of  each  watch  ] 

,         j  Gold  watch,  $80. 
janS'    \  Silver     "       $30. 

17.  There  is  a  certain  number  expressed  by  two  figures, 
which  figures  are  called  digits.  The  sum  of  the  digits  is 
11,  and  if  13  be  added  to  the  first  digit  the  sum  will  be  three 
times  the  second :  what  is  the  number  1  Ans.  56. 

18.  From  a  company  of  ladies  and  gentlemen  15  ladies 

retire ;    there  are  then  left  two    gentlemen  to   each  lady. 

After  which,  45  gentlemen   depart,  when  there  are  left  5 

ladies  to  each  gentleman :  how  many  were  there  of  each  at 

first1?  a        j  50  gentlemen. 

\  40  ladies. 

19.  A  person  wishes  to  dispose  of  his  horse  by  lottery. 
If  he  sells  the  tickets  at  $2  each,  he  will  lose  $30  on  his 
horse ;  but  if  he  sells  them  at  $3  each,  he  will  receive  $30 
more  than  his  horse  cost  him.  What  is  the  value  of  the 
horse  and  number  of  tickets  1         ,        (Horse,  $150. 

'  1  No.  of  tickets,     60. 


EQUATIONS     0  1-     THE     FIRST     DEGREE.         133 

20.  A  person  purchases  a  lot  of  wheat  at  $1,  and  a  lot  of 
rye  at  75  cents  per  bushel,  the  whole  costing  him  $117,50. 
He  then  sells  ^  of  his  wheat  and  ^  of  his  rye  at  the  same 
rate,  and  realizes  $27,50.     How  much  did  he  buy  of  each  1 

80  bush,  of  wheat. 

50  bush,  of  rye. 


Ans.  -j 


Equations  involving  three  or  more  unknown  quantities. 

77.  Lefe-  us  now  consider  equations  involving  three  or 
more  unknown  quantities. 
Take  the  equations 

5  x  —  6y  +  42  =  15, 
7x  +  4y  —  3z  =  10, 
2x  +    y  +  6z  =  4G. 

To  eliminate  z  by  means  of  the  first  two  equations,  mul- 
tiply the  first  by  3  and  the  second  by  4 ;  then,  since  the 
co-efficients  of  z  have  contrary  signs,  add  the  two  results 
together.     This  gives  a  new  equation : 

43z  —  2y  =  121. 

Multiplying  the  second  equation  by  2,  (a  factor  of  the 
co-efficient  of  z  in  the  third  equation,)  and  adding  the  result 
with  the  third  equation,  we  have 

lGx  +  9y  =  84. 

The  question  is  then  reduced  to  finding  the  values  of  z 
and  y,  which  will  satisfy  these  new  equations. 

Now,  if  the  first  be  multiplied  by  9,  the  second  by  2,  and 
the  results  added  together,  we  find 

410x  «=  1257,     whence     x  =  3. 


134  ELEMEVTART      A  L  G  E  B  B  A  . 

We  might,  by  means  of  the  two  equations  involving  x 
and  y,  determine  y  in  tne  same  way  that  we  have  deter- 
mined x ;  but  the  value  of  y  may  be  determined  more 
simply,  by  observing,  that  the  last  of  these  two  equations 
becomes,  by  substituting  for  x  its  value  found  above, 

84  —  48 
48  +  9y  =  84,    whence   y  = - =  4. 

In  the  same  manner,  the  first  of  the  three  proposed  equa 
tions  becomes,  by  substituting  the  values  of  x  and  y," 

24 
15  —  24  +  A.z  —  15,    whence   z  =  —  =.6. 

4 

Hence,  to  solve  equations  containing  three  or  more  un- 
known -quantities,  we  have  the  following 

RULE. 

I.  Eliminate  one  of  the  unknown  quantities  by  combining 
any  one  of  the  equations  with  each  of  the  others  ;  there  will 
thus  be  obtained  a  series  of  new  equations  containing  one  less 
unknown  quantity. 

II.  Eliminate  another  unknown  quantity  by  combining  one 
of  these  neto  equations  with  each  of  the  others. 

III.  Continue  this  series  of  operations  tmtil  a  single  equa 
lion  containing  but  one  unknown  quantity  is  obtained,  from 
which  the  value  of  this  unknown  quantity  is  easily  found. 
Then,  by  going  back  tlviough  the  series  of  equations  that  have 
been  obtained,  the  values  of  the  other  unknown  quantities  may 
be  successively  determined. 

77.  Give  the  general  rule  for  solving  equations  involving  three  or 
more  unknown  quantities  ?  What  is  the  first  step  ?  What  the  second  I 
What  the  thitd  I 


EQUATIONS     OF     THE     FIKBT     B  E  U  K  E  E  .         135 

78.  Remark. — It  often  happens  that  each  of  the  proposed 
equations  does  not  contain  all  the  unknown  quantities.  In 
this  case,  with  a  little  address,  the  elimination  is  very 
quickly  performed. 

Take  the  four  equations  involving  four  unknown  quanti- 
ties : 

(1.)  2x  -  2»y  +  2z  =  13.  (3.)  Ay  +  2z  —  14. 

(2.)  Au  —  2x  =  30.  (4.)  5y  +  3m  =  32. 

By  inspecting  these  equations,  we  see  that  the  elimination 
of  z  in  the  two  equations,  (1)  and  (3),  will  give  an  equation 
involving  x  and  y  ;  and  if  we  eliminate  u  in  the  equations 
(2)  and  (4),  we  shall  obtain  a  second  equation,  involving  x 
and  y.  These  last  two  unknown  quantities  may  therefore 
be  easily  determined.  In  the  first  place,  the  elimination  of 
z  from  (1)  and  (3),  gives 

ly  -  2x  =  1  ; 
That  of  u  from  (2)  and  (4),  gives 
20y  +  6x  =  38. 
Multiplying  the  first  of  these  equations  by  3,  and  adding, 

41y=41; 
Whence  y  =    1. 

Substituting  this  value  in  7y  —  2x  =  1 ,  we  find 

x  =  3. 
Substituting  for  x  its  value  in  equation  (2),  it  becomes 

4tt  -  G  =  30  : 
Whence  u  =  0. 

And  substituting  for  y  its  value  in  equation  (3),  there 
results 


130 


ELEMENTARY     ALUEBRA. 


EXAMPLES. 


1.  Given  < 


x  +      y  +      S  =  29  "^ 
x  +    2y  +    3s  =  62 


►  to  find  a;,  y  and  2. 


2.  Given 


3.  Given  - 


Ans.  x  =  8,  y  —  9,  z  —  12. 

2a;  +    4y  —    3s  =  22  "1 

4a;  —    2y  +    5s  =  18   j-  to  find  a;,  y  and  2. 

6.r  +    7y  -      2  =  63  J 

^4»s.  a;  =  3,  y  —  7,  s  =  4. 

1  1 

X  +  ~2y  +  "3  s  =  82 

—  a;  +  -rV  H z  =  15   f  to  find  a;,  y  and  s. 


-y 

O  t  t» 

1    J_ x    .  1 


12 


.4ns.  x  =  12,  y  =  20,  s  =  30. 

4.  Divide  the  number  90  into  four  such  parts  that  the 
first  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  shall  be  equal 
each  to  each. 

This  problem  may  be  easily  solved  by  introducing  a  new 
unknown  quantity. 

Let  x,  y,  z,  and  u,  be  the  required  parts,  and  designate  by 
m  the  several  equal  quantities  which  arise  from  the  condi- 
tions.    We  shall  then  have 


x  4-  2  =  m,    y  —  2  =  mt    2z  =  m,     — 


m. 


EQUATIONS     OF     THE     FIRST     DEGREE.         IIP 

From  which  we  find, 

m 

x  =m  —  2,  y  =  m-{-2,  z  =  — ,    u  =  2m. 

And  by  adding  the  equations, 

x-{-y-\-z-]-u=:m-\-m-\-  —  +  2m  =  4  £m. 

At 

And  since,  by  the  conditions  of  the  problem,  the  first 
member  is  equal  to  90,  we  have 

4±m  =  90     or,     |m  =  90  ; 

hence,  m  =  20. 

Having  the  value  of  7n,  we  easily  find  the  other  values ; 
viz. 

x  =  18,     y  =  22,     e=  10,     u  ==  40. 

5.  There  are  three  ingots  composed  of  different  metals 
mixed  together.  A  pound  of  the  first  contains  7  ounces  of 
silver,  3  ounces  of  copper,  and  6  of  pewter.  A  pound  of 
the  second  contains  12  ounces  of  silver,  3  ounces  of  copper, 
and  1  of  pewter.  A  pound  of  the  third  contains  4  ounces 
of  silver,  7  ounces  of  copper,  and  5  of  pewter.  It  is  re- 
quired to  find  how  much  it  will  take  of  each  of  the  three 
'ngots  to  form  a  fourth,  which  shall  contain  in  a  pound,  8 
ounces  of  silver,  3-|  of  copper,  and  4-j  of  pewter. 

Let  z,  y,  and  z  represent  the  number  of  ounces  which  it 
is  necessary  to  take  from  the  three  ingots  respectively,  in 
order  to  form  a  pound  of  the  required  ingot.  Since  there 
are  7  ounces  of  silver  in  a  pound,  or  16  ounces,  of  the  first 
ingot,  it  follows  that  one  ounce  of  it  contains  -j7^  of  an 
ounce  of  silver,  and   consequently   in   a  number  of  ounces 

denoted  by  x>  U-ere  is  —  ounces   of  silver.      In  the   same 


138  ELEMENTARY     ALGEBRA, 

12y  4z 

manner  we  would  find  that  ——■  and  — ,  express  the  num« 

ber  of  ounces  of  silver  taken  from  the  second  and  third,  to 
form  the  fourth ;  but  from  the  enunciation,  one  pound  of 
this  fourth  ingot  contains  8  ounces  of  silver.  We  have, 
then,  for  the  first  equation, 

7x      12y       4s 

Tg  +  "uT  +  Tg=8' 

or,  making  the  denominators  disappear, 

7x  +  12y  +  4z  =  128. 
As  respects  the  copper,  we  should  find 

3x  +  Sy  +  1z  =  60, 
and  with  reference  to  the  pewter 

6x  4-  y  +  5z  =  G8. 

As  the  co-efficients  of  y  in  these  three  equations,  are  the 
most  simple,  it  is  convenient  to  eliminate  this  unknown 
quantity  first. 

Multiplying  the  second  equation  by  4,  and  subtracting  the 
first  from  it,  member  from  member,  we  have 

5£  +  24s=:  112. 

Multiplying  the  third  equation  by  3,  and  subtracting  the 
second  from  the  resulting  equation,  we  have 

15a:  +  Sz  =  144. 

Multiplying  this  last  equation  by  3,  and  subtracting  thp 
preceding  one  from  the  resulting  equation,  Ave  obtain 

40*  =  320, 

whence  %  —  S, 


EQUATIONS     OF     THE     FIRST     DEGREE.         109 

Substitute  this  value  for  x  in  the  equation, 
15.Z  +  82  =  144  ; 
it  becomes  120  +  8s  =  144, 

whence  z  =  3. 

Lastly,  the  two  values  x  =  8,  z  =  3,  being  substituted  iu 
the  equation 

6x  -\-y  +  5z  =  68, 
give  48  +  y  +  15  =  68, 

whence  y  =  5. 

Therefore,  in  order  to  form  a  pound  of  the  fourth  ingot, 
we  must  take  8  ounces  of  the  first,  5  ounces  of  the  second, 
and  3  of  the  third. 

Verification. 

If  there  be  7  ounces  of  silver  in  16  ounces  of  the  first 
ingot,  in  8  ounces  of  it,  there  should  be  a  number  of  ounces 
of  silver  expressed  by 

7X8 
16 
In  like  manner, 

12  X  5  ,     4X3 

-16-    and     ~W 
will  express  the  quantity  of  silver  contained  in  5  ounces  of 
the  second  ingot,  and  3  ounces  of  the  third. 
Now,  we  have 

7x8       12  X  5       4x3  _  128  _ 

i6  +    To-  +  n^r  ~  To-  ~  ; 

therefore,  a  pound  of  the  fourth  ingot  contains  8  ounces  of 
silver,  as  required  by  the  enunciation.  The  same  conditions 
may  be  verified  with  respect  to  the  copper  and  pewter. 


140  ELEMENTARY      ALGEBRA. 

G.  A's  age  is  double  B's,  and  B's  is  triple  of  C's,  and  the 
sum  of  all  their  ages  is  140.     What  is  the  age  of  each  % 

Ans.  A's  =  84,  B's  =;  42,  and  C's  —  14. 

7.  A  person  bought  a  chaise,  horse,  and  harness,  for  £60 ; 
the  horse  came  to  twice  the  price  of  the  harness,  and  the 
chaise  to  twice  the  price  of  the  horse  and  harness.  What 
did  he  give  for  each  1  (  £13     6s.  &d.  for  the  horse. 

Ans.  \  £  G  18s.  4d.  for  the  harness. 
(  £40  for  the  chaise. 

8.  To  divide  the  number  36  into  three  such  parts  that  ^ 
of  the  first,  -g-  of  the  second,  and  ^  of  the  third,  may  be  all 
equal  to  each  other.  Ans.  8,  12,  and  16. 

9.  If  A  and  B  together  can  do  a  piece  of  wrork  in  8  days, 
A  and  C  together  in  9  days,  and  B  and  C  in  ten  days ;  how 
many  days  would  it  take  each  to  perform  the  same  work 
alone  ?  Ans.  A  14f|,  B  17ff,  C  23^. 

10.  Three  persons,  A,  B,  and  C,  begin  to  play  together, 
having  among  them  all  $600.  At  the  end  of  the  first  game 
A  has  won  one-half  of  B's  money,  which,  added  to  his  own, 
makes  double  the  amount  B  had  at  first.  In  the  second 
game,  A  loses  and  B  wins  just  as  much  as  C  had  at  the 
beginning,  when  A  leaves  off  with  exactly  what  he  had  at 
first.     How  much  had  each  at  the  beginning? 

Ans.  A  $300,  B  $200,  C  $100. 

11.  Three  persons,  A,  B,  and  C,  together  possess  $3640. 
If  B  gives  A  $400  of  his  money,  then  A  will  have  $320 
more  than  B;  but  if  B  takes  $140  of  C's  money,  then  B 
and  C  will  have  equal  sums.     How  much  has  each  ? 

^Ins.  A  $800,  B  $1280,  C  $1560. 

12.  Three  persons  have  a  bill  to  pay,  which  neither 
alone  is  able  to  discharge.  A  says  to  B,  "  Give  me  the 
4th  of  your  money,  and  then  I  can  pay  the  bill."  B  says 
to  C,  "  Give   me  the  8th  of  yours,  and  I  can  pay  it.     But 


Ans. 


EQUATIONS     OF     THE     FIRST     DEGREE.  141 

C  says  to  A,  "  You  must  give  me  the  half  of  yours  before 
I  can  pay  it,  as  I  have  but  $8."  What  was  the  amount  of 
their  bill,  and  how  much  money  had  A  and  B  % 

Amount  of  the  bill,  $13 
A  had  $10,  and  B  $12. 

13.  A  person  possessed  a  certain  capital,  which  he  placed 
out  at  a  certain  interest.  Another  person,  who  possessed 
10000  dollars  more  than  the  first,  and  who  put  out  his  capi- 
tal 1  per  cent,  more  advantageously,  had  an  income  greater 
by  800  dollars.  A  third  person,  who  possessed  15000  dol- 
lars more  than  the  first,  putting  out  his  capital  2  per  cent, 
more  advantageously,  had  an  income  greater  by  1500  dol- 
lars. Required  the  capitals  of  the  three  persons,  and  the 
rates  of  interest. 

(  Sums  at  interest,    $30000,     $40000,     45000. 
(  Kates  oi  interest,  4  5  0  pr.  ct. 

14.  A  widow  receives  an  estate  of  $15000  from  her  de- 
ceased husband,  with  directions  to  divide  it  among  two  sons 
and  three  daughters,  so  that  each  son  may  receive  twice  as 
much  as  each  daughter,  and  she  herself  to  receive  $1000 
more  than  all  the  children  together.  What  was  her  share, 
and  what  the  share  of  each  child  1 

f  The  widow's  share,   $8000. 

Ans.    }  Each  son's,  2000. 

(  Each  daughter's,         1000. 

15.  A  certain  sum  of  money  is  to  be  divided  between 
three  persons,  A,  B,  and  C.  A  is  to  receive  $3000  less 
than  half  of  it,  B  $1000  less  than  one-third  part,  and  C  to 
receive  $800  more  than  the  fourth  part  of  the  whole.  What 
is  the  sum  to  be  divided,  and  what  does  each  receive  ? 

r  Sum,  $38400. 

J   A  receives     16200. 

'•  11800. 

I C        "  10400 


Am.<>  B 


142  ELEMENTARY     ALGEBRA. 


CHAPTER  IV. 

Of  Powers. 

79.  If  a  quantity  be  multiplied  any  number  of  times  by 
itself,  the  product  is  called  a  power  of  the  quantity.     Thus, 

a  =  a1  is  a  root,  or  first  power  of  a. 

a  X  «  =  a2  is  the  square,  or  second  power  of  a. 

a  x  a  x  a  =  a3  is  the  cube,  or  third  power  of  a. 

axaxaxa  =  ai  is  the  fourth  power  of  a. 

axaxaxaxa  =  a5  is  the  fifth  power  of  a. 

In  every  power  there  are  three  things  to  be  considered  : 

1st.  The  quantity  which  is  multiplied  by  itself,  and  which 
is  called  the  root,  or  the  first  power. 

2d.  The  small  figure  which  is  placed  at  the  right,  and  a 
little  above  the  letter.  This  figure  is  called  the  exponent 
of  the  power,  and  shows  how  many  times  the  letter  enters 
as  a  factor. 

3d.  The  power  itself,  which  is  the  final  product,  or  result 
of  the  multiplications. 

"79.  If  a  quantity  be  continually  multiplied  by  itself,  what  is  the  pro- 
duct called  ?  How  many  things  are  to  be  considered  in  every  power  I 
What  are  thev 


OF    POWERS. 

For  example,  if  we  suppose   a  =  3,    we  have 

a=      3  the  1st  power  of  3. 

c2=r32  =  3x3=      9  the  2d  power  of  3. 

a3  =  33  =  3    x  3  x  3  =    27  the  3d  power  of  3. 

a4  =  3*  =  3    X  3    X  3  X  3  =    81   the  4th  power  of  3. 

a5  =  35  =  3    X  3    X  3    X  3  X  3  =  243  the  5th  power  of  3. 

In  these  expressions,  3  is  the  root,  1,  2,  3,  4  and  5  are 
the  exponents,  and  3,  9,  27,  81  and  243  are  the  powers. 

To  raise  a  monomial  to  any  'power. 

80.  Let  it  be  required  to  raise  the  monomial  2a362  to 
the  fourth  power.     We  have 

(2a?h2Y  =  2a3b2  X  2a362  X  2a3£2  X  2a3h% 

which  merely  expresses  that  the  fourth  power  is  equal  to 
the  product  which  arises  from  taking  the  quantity  four 
times  as  a  factor.  By  the  rules  for  multiplication,  this  pro- 
duct is 

(2a362)4  =  2V-i-3+3  +  362  +  2  +  2  +  2  =  24a1268; 

from  which  we  see, 

1st.  That  the  co-efficient  2  must  be  raised  to  the  4th 
power ;    and, 

2d.  That  the  exponent  of  each  letter  must  be  multiplied 
by  4,  the  exponent  of  the  power. 

As  the  same  reasoning  would  apply  to  every  example, 
we  have,  for  the  raising  of  monomials  to  any  power,  the 
following; 


144  ELEMENTARY     ALGEBRA. 

ETJLE. 

J.  liaise  the  co-efficient  to  the  required  power. 
II.  Multiply  the  exponent  of  each  letter  by  the  exponent  cf 
the  'power. 

EXAMPLES. 

1.  What  is  the  square  of  8a2?/3?  Ans.  9a*y% 

2.  What  is  the  cube  of  Qa!*y2x  1         ■         Ans.  21  Qal5y6x> 

3.  What  is  the  fourth  power  of  2a3y3bb  1 

Ans.  16al2yl2b2i 

4  What  is  the  square  of  a2b5y3 1  Ans.  a4bwyf 

5.  What  is  the  seventh  power  of  a2bcd3 1 

Ans.  aub1c1dz'1 

(*>.   What  is  the  sixth  power  of  a2b3c2d%       Ans.  al2bl8cl2d- 

7.  What  is  the  square  and  cube  of  —  2a2b2  ? 

Square.  Cube. 

—  2a?b2  —  2a2b2 

—  2a2b2  —  2a2P 
+  4a4Z>4.                                                     -f  4a46* 

—  2a262 


8aeb6. 


By  observing  the  way  in  which  the  powers  are  formed, 
we  may  conclude, 

1st.   When  the  root  is  %>ositive,  all  the  powers  will  be  positive. 

2d.  When  the  root  is  negative,  all  powers  denoted  by  an 
even  exponent  will  be  positive,  and  all  denoted  by  an  odd  ex- 
ponent will  be  negative. 

80.  What  is  a  monomial  ?  Give  the  rule  for  raising  a  monomial  to  any 
power.  When  the  root  is  positive,  how  will  the  powers  be  ?  When  the 
root  is  negative,  how  will  the  powers  be  I 


OP    POWERS.  145 

8.  What  is  the  square  of   —  2  a*b5  %  Ans.  4a8610 

9.  What  is  the  cube  of   —  5aby2c  %    Ans.   —  125a15y6c3. 

10.  What  is  the  eighth  power  of   —  a3xy2  1 

Ans.    +  a2ixBy16. 

11.  What  is  the  seventh  power  of   —  a2yx2  1 

Ans.    —  auy7a;14 

12.  What  is  the  sixth  power  of  2abey5  1 

Ans.  G4a6636y30 

13.  What  is  the  ninth  power  of   —  cdx2y3  1 

Ans.    —  c9d9xisy21. 

14.  What  is  the  sixth  power  of   —  3ab2d1 

Ans.  729a6W. 

15.  What  is  the  square  of  —  10a2b2c3 1     Ans.  IQQaWc6. 

16.  What  is  the  cube  of  —  9aeb5d3/2 1 

Ans.    —  729a18615d9/6. 

17.  What  is  the  fourth  power  of  —  A.abb3c*d&  ] 

Ans.  256a20612c16cf20. 

18.  What  is  the  cube  of  —  Aa2b2c3d% 

Ans.    —  G4a6b6c*d3. 

19.  What  is  the  fifth  power  of  2a3b2xy  % 

Ans.  32a15b™x5ys. 

20.  What  is  the  square  of  20x*y'ic5  %         Ans.  400xsysc10. 

21.  What  is  the  fourth  power  of  oa2b2c3 1 

Ans.  81a  W'2. 

22.  What  is  the  fifth  power  of   —  c2d3x2y2  % 

Ans.    —  c™d15x10y™. 

23.  What  is  the  sixth  power  of   —  ac2d/1 

Ans.  aBcnd6/a, 

24.  What  is  the  fourth  power  of   —  2a2c2d3 1 

Ans.  lGaW12. 
7 


146  ELEMENTARY     ALGEBRA. 


To  raise  a  polynomial  to  any  power. 

81.  A  power  of  a  polynomial,  like  that  of  a  monomial, 
is  obtained  by  multiplying  the  quantity  continually  by 
itself.  Thus,  to  find  the  fifth  power  of  the  binomial  a  -f  6, 
we  have 

a   -\-    b       .........      1st  power. 

a  +    b 
a2  +    ctb 

+    ab  +  b2 

a2  +  2ab  -f-  b2        . 2d  power. 

a  +    b 

a3  +  2a26  -j-      a&8 

-|-    a26+    2a62   +    &3 
a3  +  3a26  -f-    Sab2  +    63     ....     3d  power, 
a   +    b 
o4  +  3a3H    3a262-f-    a63 

+    a3b  +    3a262  +    3a63   +  b* 
a*  +  4a36+    6a262  +    4a63  +    64        4th  power. 
a   +   b 
a£  f  4a46+    6W  +    4a2I3  +    a&* 

+    a4&  +    4a362  -f-    6a263  -f  4a64  +  &5 
a5  +  5a*b  +  10a362  +  10a263  +  SaV  +  65     Ans. 

Remark. — 82.  It  will  be  observed  that  the  number  of 
multiplications  is  always  1  less  than  the  units  in  the  expo- 

81    LTow  is  the  power  of  a  polynomial  obtained  t 


OF     POWERS.  147 

&ent  of  the  power.  Thus,  if  the  exponent  is  1,  no  multipli- 
cation is  necessary.  If  it  is  2,  we  multiply  once  ;  if  it  is  3, 
twice ;  if  4,  three  times,  &c.  The  powers  of  polynomials 
may  be  expressed  by  means  of  an  exponent.  Thus,  to  ex- 
press that  a  -f  b  is  to  be  raised  to  the  5th  power,  we  write 

(a  -f-  6)5, 

v>  Inch  is  the  expression  for  the  fifth  power  of  a  -f-  h. 

2.  Find  the  5th  power  of  the  binomial  a  —  b. 


a   —    b 
a  —    b 

1st 

power. 

a2  —    ab 

—    ab   +  62 

a2  —  2ab   +  b2       ,     . 
a   —      6 

2d 

power. 

a3  —  2a2b  + 

ab2 

—    a2b-\- 

2ab2   — 

b3 

3d 

a3  —  3a26  -f- 

3ab2   — 

b3     .     .     .     . 

power. 

a   -    b 

4th 

a4  —  3a3&  -f- 
—    a3b  -f- 

3a262  - 
2,o?b2  - 

ab3 
Sab3   +   ¥ 

a*  —  4a36  + 

Qa2b2  - 

Aab3   +   b* 

power. 

a   —    b 

a5  —4  a4b  -f- 

6a3b2  - 

4a2b3  +    ab* 

—    a46  + 

Aa3b2  - 

Ga2b3  -f  4«6*  - 

-b5 

a5  —  5a46  +  10a3b2  - 

10a2 b3  +  5a&4  - 

-b* 

Ans. 

82.    ITow   does  the    number  of  multiplications  compare  with  th8 
exponent  of  the  power?     If  the  exponent  is  4,  how  many  multi plica' 

lions  I 


148  ELSMEJSTARY     ALGEBHi, 

3.  What  is  the  square  of    5a  ~  2c  -j-  d  ? 


5a 

-    2c    + 

d 

5a 

-    2c    + 

d 

25a2 

—  lOac  + 

5ad 

—  lOac  + 

4C2   _ 

-  2«2 

+ 

5(7(7- 

-2«J  + 

o72 

25a2  —  20ac  +  lOarf-f-  Ac2  —  4cd  +  d2     Arts. 
4.  Find  the  4th  power  of  the  binomial    3a  —  2b. 

3a   —      26        1st  power. 

3a   —      2b 


9a2  —      Gab 

—       Gab    +  462 

9a2  —  12w6   +  46* 2d  power. 

3a   —      2b 


27a3  —    36a26  +  12a62 

—  18a3&  +  24a62  —    863 

27a3  —    54a26~+  3Ga62  —    863     .     .      3d  power. 

3a    —      2b 

81a4  —  162a36  +  108a262  —  24a53 

—  54a36  +  10Sa262  —  T2ab3  +  166* 
81a4  — ~216a36  +  2lGa2b2  —  96a63  4-  166* 

5.  What  is  the  square  of  the  binomial  a  +  1? 

^?is.  a2  -h  2a  4-  1, 

6.  What  is  the  square  of  the  binomial   a  —  1  1 

A*is.  a2  —  2a  -f  1. 

7.  What  is  the  cube  of  9a  —  36  % 

Ans.  729a3  —  729a26  +  243a62  —  2761 

8.  What  is  the  third  power  of    a  —  11 

Ans.  a3  —  3a2  +  3a  -  I. 


OF     POWERS.  149 

9    "What  is  the  4th  power  of  x  —  y  1 

Ans.  x*  —  4x3y  +  6x2y2  —  4xy3  -\-  y%, 

10    What  is  the  cube  of  the  trinomial    x  +  y  -f  si 

Answer. 
x*4-3x2yi-3x2z+3xy2+3xz2  +  &j2z+3ijz2+Qxyz+y3+z3. 

11.  What  is  the  cube  of  the  trinomial   2a2  —  4ab  +  3621 
Answer. 
fia   -  48a56  +  132a462  —  208a363-|-  ISSaW  —  108a65+2766 

To  raise  a  fraction  to  any  'power. 

83.  A  power  of  a  fraction  is  obtained  by  multiplying  the 
fraction  by  itself  a  certain  number  of  times ;  that  is,  by 
multiplying  the  numerator  by  the  numerator,  and  the  deno- 
minator by  the  denominator. 

Thus,  the  cube  of   —     which   is  written 


6 


(t)3  = 


a         a         a         as 

"6~  x  T  x  T  =  T3' 


Is  found  by  cubing  the  numerator  and  denominator  sepa- 
rately. 


2. 

What  is 

>  the 

square 

of  the  fraction 

a  — 
b  + 

c 

—  1 
c 

We  have 

ir  ■ 

_(a- 
"(*  + 

c)2 
c)2 

a2  —  2ac  +  cz 
b2  +  26c  +  c2 

Ans. 

3. 

What  is 

the  cube  of 

xy 
36c 

? 

Ans. 

x3y3 

276V 

83.  IIow  do  you  find  the  power  of  a  fraction  t 


1 50  ELEMENTARY     ALGEBRA 

ab2c 


4.  What  is  the  fourth  power  of 

Ans 

5.  What  is  the  cube  of  tZ.1  ? 

*  +  y 


2x2y2  * 

lGa^y5' 


.         a;3  —  Sx2y  -f  3a;y2  —  y3 
x3  +  3x,2y  +  3a:?,'2  -f-  y3 

6.  What  is  the  fourth  power  of  —  ?         ^4?is 


4cry   "  10y4 

7.  What  is  the  fifth  power  of    1         Am. 


lSyz    '  32yV 


S.  What  is  the  square  of   r - 

by  —  x 

Arts 

b2y2  —  2bxy  -f  x2 


by  —  x  ' 

.          a2x2  —  2a  ry  4-  y2 
Ans.    —— : „,        .      „ 


9.  What  is  the  cube  of i 

x+2y- 

Sa3  —  S6a2b  4-  54o62  -  2763 
x3  +  <!>x2y  +  \2xy2  +  8y3 

Binomial  Formula. 

84.  The  method  which  has  been  explained  of  raising  a 
binomial  to  any  power,  is  somewhat  tedious,  and  hence 
other  methods,  less  difficult,  have  been  anxiously  sought 
for.  The  most  simple  which  has  yet  been  discovered,  is 
that  of  Sir  .saac  Newton,  by  means  of  the  Binomial 
Formula. 


84.  What  is  the  object  of  the  Binomial  Formula?    "Who  discovered 
this  formula  ? 


BINOMIAL    FORMULA.  151 

85.  In  raising  a  quantity  to  any  power,  it  is  plain  that 
there  are  four  things  to  be  considered : — - 

1st.  The  number  of  terms  of  the  power, 

2d.  The  signs  of  the  terms. 

3d.  The  exponents  of  the  letters. 

4th.  The  co-efficients  of  the  terms. 

Of  the  Terms. 

86.  If  we  take  the  two  examples  of  Article  81,  which 
we  there  wrought  out  in  full ;  we  have 

(a  +  b)5  =  a5  +  5a46  +  10a3Z>3  +  lQa2P  +  5a64  +  b5  ; 

(a  —  by  =  a5  —  5a46  +  lOaW  —  10a2b3  +  5a&4  —  b5. 

By  examining  the  several  multiplications,  in  Art.  81,  we 
shall  observe  that  the  second  power  of  a  binomial  contains 
three  terms,  the  third  power  four,  the  fourth  power  five,  the 
fifth  power  six,  &c. ;  and  hence,  we  may  conclude — 

That  the  number  of  terms  in  any  power  of  a  binomial,  is 
greater  by  o?ie,  than  the  exponent  of  the  potver. 

Of  the  Signs  of  the  Terms. 

87.  It  is  evident  that  when  both  terms  of  the  given 
binomial  are  plus,  all  the  terms  of  the  potver  will  be  plus. 

2d.  If  the  second  term  of  the  binomial  is  negative,  then 
all  the  odd  terms,  counted  from  the  left,  will  be  positive,  and 
all  the  even  terms  negative. 

85.  In  raising  a  quantity  to  any  power,  how  many  things  are  to  be 
considered  ?     What  are  they  ? 

86.  How  many  terms  are  there  in  any  power  of  a  binomial?  If  the 
exponent  is  3,  how  many  terms  ?     If  it  is  4,  how  many  terms  ?    If  5  ?  &c 

87.  If  both  terms  of  the  binomial  are  positive,  how  are  the  terms  of 
the  power  ?  If  the  second  term  i3  negative,  how  are  the  Eigns  of  the 
terms  f 


152  ELEMENTARY     ALGEBRA, 

Of  the  Exponents. 

88.  The  letter  which  occupies  the  first  place  in  a  bino 
mial,  is  called  the  leading  letter.  Thus,  a  is  the  leading 
letter  in  the  binomials  a  -\-  b,  a  —  b. 

1st.  It  is  evident  that  the  exponent  of  the  leading  letter, 
in  the  first  term,  will  be  the  same  as  the  exponent  of  the 
power;  and  that  this  exponent  will  diminish  by  unity  in 
each  term  to  the  right,  until  we  reach  the  last  term,  which 
does  not  contain  the  leading  letter. 

2d.  The  exponent  of  the  second  letter  is  1,  in  the  second 
term,  and  increases  by  unity  in  each  term  to  the  right  until 
we  reach  the  last  term,  in  which  the  exponent  is  the  same 
as  that  of  the  given  power. 

3d.  The  sum  of  the  exponents  of  the  two  letters,  in  any 
term,  is  equal  to  the  exponent  of  the  given  power.  This 
last  remark  will  enable  us  to  verify  any  result  obtained  by 
means  of  the  binomial  formula. 

Let  us  now  apply  these  principles  in  the  two  following 
examples,  in  which  the  co-efficients  are  omitted : — - 

(a  +  b)5  .   .   .  a6  +  a5b  +  a4b2  +  a3£3  -f  a?b4  +  abh  +  b\ 

(a  —  by  ...  a6  —  a5b  +  a*b2  -  a3b3  +  a26*  —  ab5  +  b5. 

As  the  pupil  should  be  practised  in  writing  the  terms, 
with  their  proper  signs,  without  the  co-efficients,  we  will  add 
a  few  more  examples. 

88.  Which  is  the  leading  letter  of  a  binomial  ?  What  is  the  exponent 
of  this  letter  in  the  first  term  ?  How  does  it  change  in  the  terms  to 
wards  the  right?  What  is  the  exponent  of  the  second  letter  in  the 
second  term  ?  How  does  it  change  in  the  terms  towards  the  right  S 
What  is  it  in  the  last  term  ?  What  is  the  sum  of  the  exponents  in  any 
term  eijual  to  ? 


BINOMIAL     FORMULA.  153 

1.  (a  +  b)3  ,  .  a3  +  a?b  +  ab2  +  P. 

2.  (a  —  by  .  *.  a*  —  a3i  +  aW  —  ab3  +  bK 

3.  (a  -f  6)5  .  .  a5  +  a4Z>  4-  a3A2  +  a2P  +  a&4  +  Is. 

4.  (a -J)7.  .  a7— a%  +  abP  — a*P-{-a3bi  — a2654-a&6— 6T, 

0/"  the  Co-efficients. 

89.  The  co-efficient  of  the  first  term  is  unity.  The  co 
efficient  of  the  second  terra  is  the  same  as  the  exponent  of 
the  given  power.  The  co-efficient  of  the  third  term  is  found 
by  multiplying  the  co-efficient  of  the  second  term  by  the 
exponent  of  the  leading  letter,  and  dividing  the  product  by 
2.     And  finally — 

If  the  co-efficient  of  any  term  be  multiplied  by  the  exponent 
of  the  leading  letter,  and  the  product  divided  by  the  number 
which  marks  the  place  of  that  term  from  the  left,  the  quotient 
will  be  the  co-efficient  of  the  next  term. 

Thus,  to  find  the  co-efficients  in  the  example 

(a  —  b)1  .  .  .  a1 —aeb-^-a5P—a4P+a3bi—a^5+aP  —  b^, 

we  first  place  the  exponent  7  as  a  co-efficient  of  the  second 
term.  Then,  to  find  the  co-efficient  of  the  third  term,  we 
multiply  7  by  G,  the  exponent  of  a,  and  divide  by  2.  The 
quotient  21  is  the  co-efficient  of  the  third  term.  To  find  the 
co-efficient  of  the  fourth,  we  multiply  21  by  5,  and  divide 
the  product  by  3  :  this  gives  35.  To  find  the  co-efficient  of 
the  fifth  term,  we  multiply  35  by  4,  and  divide  the  product 
by  4  :  this  gives  35.  The  co-efficient  of  the  sixth  term, 
found  in  the  same  way,  is  21  ;  that  of  the  seventh,  7  ;  and 
that  of  the  eighth,  1.     Collecting  these  co-efficients, 

(a~by  = 

0*_7a66+21a5i>2-35a463  +  35a3^-21a265-f7a56-47. 
7* 


154  ELEMENTARY     ALGEBRA. 

Remark. — We  see,  in  examining  this  last  result,  that  the 
co-efficients  of  the  extreme  terms  are  each  unity,  and  that 
the  co-efficients  of  terms  equally  distant  from  the  extreme 
terms  are  equal.  It  will,  therefore,  be  sufficient  to  find  the 
co-efficients  of  the  first  half  of  the  terms,  and  from  these 
the  others  may  be  immediately  written. 

EXAMPLES. 

1.  Find  the  fourth  power  of  a  -f  b. 

Am.  ai  -f  4a?b  +  Ga2b2  +  4ab3  -f  b\ 

2.  Find  the  fourth  power  of  a  —  b. 

Am.  a4  —  4a3b  +  6a2b2  —  4aP  -f  b\ 

3.  Find  the  fifth  power  of  a  +  b. 

Am.  a5  +  5a46  +  10a3b2  -f  10a2P  +  5«6*  +  b\ 

4.  Find  the  fifth  power  of  a  —  b. 

Am.  a5  —  5a46  +  10a3P  —  10a2b3  +  5a&4  —  6s. 

5.  Find  the  sixth  power  of  a  -f-  b. 

Am.  a6  +  Ga5b  +  \haAb2  +  20a 3b3  +  15a264  +  Gab5  +  b*. 

6.  Find  the  sixth  power  of  a  —  5. 

Ans.  a6  —  Ga5b  +  15a462  —  20a3i3  +  15a264  —  Gab5  +  b6. 

7.  Let  it  be  required  to  raise  the  binomial  3a2c  —  2bd  to 
the  fourth  power. 

It  frequently  occurs  that  the  terms  of  the  binomial  are 
affected  with  co-efficients  and   exponents,  as  in   the  above 


8'J.  What  is  the  co-efficient  of  the  first  term  ?  What  is  the  co-efficient 
of  the  second  ?  How  do  you  find  the  co-efficient  of  the  third  term  ? 
How  do  you  find  the  co-efficient  of  any  term  ?  What  are  the  co-effi- 
cients of  the  first  and  last  terms  ?  How  are  the  co-efficients  of  terma 
equally  distant  from  the  two  extremes  f 


BINOMIAL      FORMULA.  155 

example.     In  the  first  place,  we  represent  each  term  of  the 
binomial  by  a  single  letter.     Thus,  we  place 

oa2c  =  x,     and     —  2bd  =  y, 

we  then  have 

(x  +  y)*  =  #4  +  4x3y  +  6a;2?/2  +  4rry3  +  yA. 

But,       a;2  =  9a4c2,        x3  =  27afic3,        x4  =  81a8c4  ; 

and        y2  =  4b2d2,    y3  =  —  8i3c/3,        y4  =  lG&4rf*. 

Substituting  for   x   and   y   their  values,  we  have 

(3a3c-2My  =  (3a2cy-\-4(3a2cy(-2bd)  +  6(3a2c)2(-2bdy 
+  4(3a2c)  (  -  2bd)3  +  (  -  2bd)\ 

and  by  performing  the  operations  indicated, 

(3a3c  —  2bdy  =  &Usci—216aec3bd+21Ga*c2b2d2—9Ga2cb3d 
+  1  Gb*d\ 

8.  What  is  the  square  of  3a  —  6b  ? 

Ans.  9a2  —  3Ga&  4-  306^ 
0.  What  is  the  cube  of  Zx  —  Oy  1 

Ans.  27 x3  —  lG2.r2y  +  324zy2  —  21 0>y\ 

10.  What  is  the  square  of    x  —  y  % 

Ans.  x2  —  2xy  -f-  y2* 

11.  What  is  the  eighth  power  of  m  +  n% 

Ans.  m8 + 8wz7«  +  2$m6n2  +  5Gm%3  +  70m4?i* + 5G»i3»' 
+  28m2«6  +  8»m7  +  n8. 

12.  What  is  the  fourth  power  of   a  —  2>b% 

Ans.  a4  -  12a36  4-  54a2i2  -  108a&3  +  816" 

13.  What  is  the  fifth  power  of    c  —  2d  ? 

Ans.  c5  -  10c+d  +  40c3(/2  —  80c2d3  4-  80cJ4  —  32^ 

14.  What  is  the  cube  of    5a  —  od  1 

Ans.  125a3  -  225a2J  4-  135arf2  —  27cP. 


156  ELEMENTARY    ALGEBRA. 

Remark.  The  powers  of  a  polynomial  may  easily  be 
found  by  the  Binomial  Formula. 

15.  For  example,  raise   a  -j-  6  +  c   to  the  third  power. 
First,  put     .     .     .     .     6  -j-  c  =  d : 

Then,    (a  +  b  +  c)3  =  (a  +  J)3  =  a3  +  3a2J  +  3ad2  +  d3. 
Or,  by  substituting  for  the  value  of  d, 

(a  +  6  +  c)3  =  a3  +  3a26  +  3a62  +  b3 

3a2c  +  362c  +  6a  6c 

+  Sac2  +  36c2 

+      c3. 

This  expression  is  composed  of  the  sum  of  the  cubes  of  the 
three  terms,  plus  three  times  the  square  of  each  term  by  the 
product  of  the  first  powers  of  the  two  others,  plus  six  times 
the  product  of  the  three  terms.  It  is  easily  proved  that  this 
law  is  true  for  any  polynomial. 

To  apply  the  preceding  formula  to  the  development  of 
the  cube  of  a  trinomial,  in  which  the  terms  are  affected  with 
co-efficients  and  exponents,  designate  each  term  by  a  single 
letter,  then  replace  the  letters  introduced,  by  their  values,  and 
perform  the  operations  indicated. 

From  this  rule,  we  find  that 

(2a2  -  4a6  +  362)3  =  8a6  —  48a56  +  132a462  —  208a363 
+  198a264  —  108a65  +  2?66. 

The  fourth,  fifth,  dec,  powers  of  any  polynomial  can  be 
found  in  a  similar  manner. 

16.  What  is  the  cube  of  a  —  26  +  c  1 

Ans.  a3  -  863  -f  c3  -  6a26  +  3a2c  +  12a62  -f-  1262c4-3ac» 
—  66c2  —  12o6c. 


EXTRACTION  OF  THE  SQUARE  ROOT.    157 


CHAPTER  V. 

Extraction  of  the  Square  Boot  of  Numbers.  Formation 
of  the  Square  and  Extraction  of  the  Square  Root  of 
Algebraic  Quantities.  Calculus  of  Radicals  of  the 
Second  Degree. 

90.  The  square  or  second  power  of  a  number,  is  the  pro- 
duct which  arises  from  multiplying  that  number  by  itself 
once:  for  example,  49  is  the  square  of  7,  and  144  is  the 
square  of  12. 

91.  The  square  root  of  a  number  is  that  number  which, 
being  multiplied  by  itself  once,  will  produce  the  given  num- 
ber. Thus,  7  is  the  square  root  of  49,  and  12  the  square 
root  of  144  :  for  7,  X  7  =  49,  and  12  X  12  =  144. 

92.  The  square  of  a  number,  either  entire  or  fractional,  is 
easily  found,  being  always  obtained  by  multiplying  this 
number  by  itself  once.  The  extraction  of  the  square  root 
of  a  number  is,  however,  attended  with  some  difficulty,  and 

equires  particular  explanation. 


90.  What  is  the  square,  or  second  power  of  a  number  J 
l>li   What  ia  the  square  root  of  a  number? 


158  ELEMENTARY     ALGEBRA. 

The  first  ten  numbers  are, 

1,       2,       3,       4,       5,       G,       7,       8,       9,       10; 

and  their  squares, 

1,       4,       9,     1G,     25,     3G,     49,     G4,     81,    100; 

and  reciprocally,  the  numbers  of  the  first  line  are  the  square 
roots  of  the  corresponding  numbers  of  the  second.  We  may 
also  remark  that,  the  square  of  a  number  expressed  by  a  single 
figure,  will  contain  no  unit  of  a  higher  denomination  than 
tens* 

The  numbers  of  the  last  line,  1,  4,  9,  1G,  &c,  and  all 
other  numbers  which  can  be  produced  by  the  multiplication 
of  a  number  by  itself,  are  called  perfect  squares. 

It  is  obvious  that  there  are  but  nine  perfect  squares  among 
all  the  numbers  which  can  be  expressed  by  one  or  two  figures : 
the  square  roots  of  all  other  numbers,  expressed  by  one  or 
two  figures,  will  be  found  between  two  whole  numbers  dif- 
fering from  each  other  by  unity.  Thus  55,  which  is  comprised 
between  49  and  64,  has  for  its  square  root  a  number  between 
7  and  8.  Also  91,  which  is  comprised  between  81  and  100, 
has  for  its  square  root  a  number  between  9  and  10. 

93.  Every  number  may  be  regarded  as  made  up  of  a 
certain  number  of  tens  and  a  certain  number  of  units.  Thus 
64  is  made  up  of  6  tens  and  4  units,  and  may  be  expressed 
under  the  form  60  +  4. 


92.  What  will  be  the  highest  denomination  of  the  square  of  a  number 
expressed  by  a  single  figure  ?  What  are  perfect  squares  ?  How  many 
ore  there  between  1  and  100  ?     What  are  they  ? 


*  Se«  Arithmetic,  Art.  8. 


EXTRACTION     OF     THE    SQUARE     ROOT.         159 

Now,  if  we  represent  the  tens  by  a  and  the  units  by  6, 
we  shall  have 

a  +  b    =  64, 
and  (a  +  6)2=(64)2; 

or  a2  +  2ab  +  b2  =  4096. 

Which  proves  that  the  square  of  a  number  composed  of 
tens  and  units,  equals  the  square  of  the  tens  jjIus  twice  the 
product  of  the  tens  by  the  units,  plus  the  square  of  the  units. 

94.  If  now,  we  make  the  units  1,  2,  3,  4,  &c,  tens,  or 
units  of  the  second  order,  by  annexing  to  each  figure  a 
cipher,  we  shall  have 

10,     20,     30,     40,     50,     00,     70,     80,     90,     100, 

and  for  their  squares, 

100,  400,  900,  1600,  2500,  3000,  4900,  6400,  8100,  10000. 

From  which  we  see  that  the  square  of  one  ten  is  100,  the 
square  of  two  tens  400  ;  and  generally  (hat  the  square  of 
tens  will  contain  no  unit  of  a  less  denomination  than  hun- 
dreds, nor  of  a  hie/her  name  than  thousands. 

Ex.  1.— To  extract  the  square  root  of  6084. 

Since  this  number  is  composed  of  more  than 
two    places    of   figures,  its    root    will    contain  60  84 

more,    than    one.      But   since    it    is    less    than 
10000,  which  is  the  square  of  100,  the  root  will  contain  but 
two  figures:  that  is,  units  and  tens. 

Now,  the  square  of  the  tens  must  be  found   in  the  two 


93.  How  may  every  number  be  regarded  as  made  iid  ?  What  is  the 
square  of  a  number  composed  of  tens  and  units  equal  to  S 

94  What  ia  the  equare  of  one  ten  equal  to?  Of  2  tone*  Of  3 
tens?  «fec 


1G0  ELEMENTARY     ALGEBRA. 

left-hand  figures,  which  we  will  separate  from  the  other  two 
by. putting  a  point  over  the  place  of  units,  and  a  second  over 
the  place  of  hundreds.  These  parts,  of  two  figures  each,  are 
called  periods.  The  part  60  is  comprised  between  the  two 
squares  49  and  64,  of  which  the  roots  are  7  and  8  :  hence, 
7  expresses  the  number  of  tens  sought ;  and  the  required  root 
is  composed  of  7  tens  and  a  certain  number  of  units. 
The  figure  7  being  found,  we 


write  it  on  the  right  of  the  given  60  84 

number,  from  which  we  separate  49 

it  by  a  vertical  line  :    then  we     7  x  2  =  14 
subtract  its  square,  49,  from  60, 


78 


118  4 
118  4 

which  leaves  a  remainder  of  11,  0 

to  which  we  bring  down  the  two 

next  figures  84.  The  result  of  this  operation,  1184,  con- 
tains twice  the  product  of  the  tens  by  the  units,  plus  the  square 
of  the  units. 

But  since  tens  multiplied  by  units  cannot  give  a  product 
of  a  less  unit  than  tens,  it  follows  that  the  last  figure,  4, 
can  form  no  part  of  the  double  product  of  the  tens  by  the 
units  :  this  double  product  is  therefore  found  in  the  part  118, 
which  we  separate  from  the  units'  place,  4. 

Now  if  we  double  the  tens,  which  gives  14,  and  then  divide 
118  by  14,  the  quotient  8  will  egress  the  units,  or  a  num- 
ber greater  than  the  units.  This  quotient  can  never  be  too 
small,  since  the  part  118  will  be  at  least  equal  to  twice  the 
product  of  the  tens  by  the  units:  but  it  may  be  too  large  ; 
for  the  118,  besides  the  double  product  of  the  tens  by  the 
units,  may  likewise  contain  tens  arising  -from  the  square 
of  the  units.  To  ascertain  if  the  quotient  8  expresses  the 
number  of  units,  we  write  the  8  on  the  right  of  the  14, 
which  gives  148,  and  then  we  multiply  148  by  8.  Thus, 
wc  evidently  form,  1st,  the  square  of  the  units  ;  and, 
2d,   the  double   product   of  the  tens  by  the   units,     This 


EXTRACTION  OF  THE  SQUARE  ROOT.    161 

jiultiplieation  being  effected,  gives  for  a  product  1184,  a 
number  equal  to  the  result  of  the  first  operation.  Having 
subtracted  the  product,  we  find  the  remainder  equal  to  0 : 
bence,78  is  the  root  required. 

Indeed,  in  the  operations,  we  have  merely  subtracted 
from  the  given  number  6084,  1st,  the  square  of  7  tens,  or  of 
70  ;  2d,  twice  the  product  of  70  by  8  ;  and,  3d,  the  square 
of  8 :  that  is,  the  three  parts  which  enter  into  the  composi- 
tion of  the  square  70  +  8,  or  78 ;  and  since  the  result  of 
the  subtraction  is  0,  it  follows  that  78  is  the  square  root  of 
60S4. 

95.  Remark. — The  operations  in  the  last  example  have 
been  performed  on  but  two  periods,  but  it  is  plain  that  the 
same  methods  of  reasoning  are  equally  applicable  to  larger 
numbers,  for  by  changing  the  order  of  the  units,  we  do  not 
change  the  relation  in  which  they  stand  to  each  other. 

Thus,  in  the  number  60  84  95,  the  two  periods  60  84 
have  the  same  relation  to  each  other  as  in  the  number 
60  84 ;  and  hence  the  methods  used  in  the  last  example 
are  equally  applicable  to  larger  numbers. 

96.  Hence,  for  the  extraction  of  the  square  root  of 
numbers,  we  have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures 
each,  beginning  at  the  right  hand: — the  period  on  the  left  will 
often  contain  but  one  figure. 

II.  Find,  the  greatest  square  in  the  first  period  on  the  left, 
and  place  its  root  on  the  right,  after  the  manner  of  a  quotient 


95.  Will  the  reasoning  in  the  example  apply  to  more  than  two 

criocta  ? 

8 


102  ELEMENTARY     ALGEBRA. 

in  division.  Subtract  the  square  of  this  root  from  the  first 
period,  and  to  the  remainder  bring  down  the  second  period  for 
a  dividend. 

III.  Double  the  root  already  found,  and  place  it  on  the  left 
for  a  divisor.  Seek  how  many  times  the  divisor  is  contained 
in  the  dividend,  exclusive  of  the  right-hand  figure,  and  .place 
the  figure  in  the  root  and  also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor,  thus  augmented,  by  the  last  figure 
of  the  root,  and  subtract  the  product  from  the  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  dividend. 
But  if  any  of  the  products  should  be  greater  tiian  the  divi- 
dend, diminish  the  last  figure  of  the  root  by  one. 

V.  Double  the  whole  root  already  found,  for  a  new  divisor 
and  continue  the  operation  as  before,  until  all  the  periods  an 
brought  down. 

97.  1st.  Remark. — If,  after  all  the  periods  are  brought 
down,  there  is  no  remainder,  the  proposed  number  is  a  per- 
fect square.  But  if  there  is  a  remainder,  you  have  only 
found  the  root  of  the  greatest  perfect  square  contained  in 
the  given  number,  or  the  entire  part  of  the  root  sought. 

For  example,  if  it  were  required  to  extract  the  square 
root  of  665,  we  should  find.  25  for  the  entire  part  of  the 
root,  and  a  remainder  of  40,  which  shows  that  665  is  not 
a  perfect  square.  But  is  the  square  of  25  the  greatest  per- 
fect square  contained  in  665  1  that  is,  is  25  the  entire  part 
of  the  root?  To  prove  this,  we  will  first  show  thj>t,  the 
difference  between  the  squares  of  two  consecutive  numbers,  is 
equal  to  twice  the  less  number  augmentel  by  one. 

96.  Give  the  rule  for  extracting  the  square  root  of  numbers.  What  is 
the  first  step  ?  What  the  second  ?  What  the  third  i  What  the  fourth  f 
What  the  fifth! 


EXTRACTION  OF  THE  SQUARE  ROOT.    1G3 

Let  .  a  =     the  less  number, 

and  .  .  a  +  1      =     the  greater. 

Then  .  («  +  l)2  =  a2  +  2a  +  1, 

and  .  .  (a)2  =  a2. 

Their  difference  =       2a  +  1     as  enunciated. 

Hence,  the  entire  part  of  the  root  cannot  be  augmented 
unless  the  remainder  is  equal  to  or  greater  than  twice  the 
root  found,  plus  one. 

But  25  X  2  +  1  =  51  >  40  the  remainder:  therefore, 25 
is  the  entire  part  of  the  root. 

98.  2d  Remark. — The  number  of  places  of  figures  in  the 
root  will  always  be  equal  to  the  number  of  periods  into 
which  the  given  number  is  separated. 


EXAMPLES. 

1.  To  find  the  square  root  of  7225.  Ans.  85 

2.  To  find  the  square  root  of  17G89.  Ans.  133. 

3.  To  find  the  square  root  of  994009.  Ans.  997. 

4.  To  find  the  square  root  of  85G73536.  Ans.  9256. 

5.  To  find  the  square  root  of  67798756.  Ans.  8234. 

6.  To  find  the  square  root  of  978121.  Ans.  989. 

7.  To  find  the  square  root  of  956484.  Ans.  978. 

8.  What  is  the  square  root  of  36372961  1  Ans.  6031. 

9.  What  is  the  square  root  of  22071204]  Ans.  4698. 

10.  What  is  the  square  root  of  106929  1  Ans.  327. 

11.  What  is  the  square  root  of  12088808379025  ? 

Ans.  3476905. 


How  many  figures  will  you  always  find  in  the  root  t 


104 


ELEMENTARY     ALGEBRA. 


99.  3d  Remark. — If  the  given  number  has  not  an  exact 
root.,  there  wiJl  be  a  remainder  after  all  the  periods  are 
brought  down,  in  which  case  ciphers  may  be  annexed,  funn- 
ing new  periods,  for  each  of  which  there  will  be  one  deci- 
mal place  in  the  root. 

1.  What  is  the  square  root  of  3GT29  1 


In  this  example  there  are 
two  periods  of  decimals, 
and  hence,  two  places  of 
decimals  in  the  root. 


3  67  29  191 
1 

.64  + 

2  9  267 
|261 

38  1 

329 
381 

382  6 

1 

24800 
22956 

3S32  4 

t 184400 
153296 

31104 

Rem. 

2.  What  is  the  square  root  of  2268741  \ 

3.  What  is  the  square  root  of  7596796  ? 

4.  What  is  the  square  root  of  96  % 

5.  What  is  the  square  root  of  153  ? 
C.  What  is  the  square  root  of  101 1 


Ans.  1506.23  +. 

Ans.  2756.22  -f . 

Ans.  9.79795  +. 
Ans.  12.36931  +. 
Ans.  10.04987  +. 


P9i  llow  will  you  find  tho  decimal  part  of  the  root  I 


EXTRACTION     OF     THE     SQUARE     ROOT         165 

7.  What  is  the  square  root  of  28597039GG44  1 

Ans.  5347G2. 

8.  What  is  the  square  root  of  41G05S00G25  % 

Ans.  203975. 

9.  What  is  the  square  root  of  483035S420G084  ? 

Ans.  G950078. 

Extraction  of  the  square  root  of  Fractions. 

100.  Since  the  square  or  second  power  of  a  fraction  is 
obtained  by  squaring  the  numerator  and  denominator  sepa- 
rately, it  follows  that  the  square  root  of  a  fraction  will  be 
equal  to  the  square  root  of  the  numerator  divided  by  the 
square  root  of  the  denominator. 

CL  (X 

For  example,  the  square  root  of    —    is  equal  to   — :  foi 

a         a         a2 
T  x  T~~¥' 

1.  What  is  the  square  root  of    —  ? 

4 

9 

2.  What  is  the  square  root  of   —1 

*  1G 

PA 

3.  What  is  the  square  root  of    — —  ? 

81 

25G 

4.  What  is  the  square  root  of    - — -? 

*  3G1 

5.  What  is  the  square  root  of     —  ? 

H  G4 


100.  If  the    numerator   and   denominator  of  a  fraction   are   perfect 
squares,  how  will  you  extract  the  square  root  ? 


Ans. 

1 
2' 

Ans. 

3 
4' 

Ans. 

8 

9* 

Ans. 

1G 
19' 

Ans. 

1 
IT" 

160  ELEMENTARY     ALGEBRA. 

4000  04 

6.  What  is  the  square  root  of    — — : —  ?  Ans. . 

1  61000  247 

r,    xktx,  ..  •   *u  .    582169  .  .        763 

7.  \V  hat  is  the  square  root  or ?  Ans. 

101.  If  neither  the  numerator  nor  the  denominator  is  a 
perfect  square,  the  root  of  the  fraction  cannot  be  exactly 
found.  We  can,  however,  easily  find  the  approximate  root. 
For  this  purpose, 

Multiply  both  terms  of  the  fraction  by  the  denominator, 
which  makes  the  denominator  a  perfect  square  without  altering 
the  value  of  the  fraction.  Then,  extract  the  square  root  of  the 
numerator,  and  divide  this  root  by  the  root  of  the  denomina- 
tor ;  this  quotient  will  be  the  approximate  root. 

g 

Thus,  if  it  be  required  to  extract  the  square  root  of  — > 

o 

15 

we  multiply  both  terms  by  5,  which  gives     —  • 

We  then  have 

-/T5  =  3.8720  +  : 

hence,  3.8720  +  —  5  =  .7745   +  =  Ans. 

7 
2.  What  is  the  square  root  of    —  1         Ans.  1.32287  +. 


14 

3.  What  is  the  square  root  of    —  %       Ans.  1.24721  +. 

4.  What  is  the  square  root  of    11—'? 

Id 

■Ans.  3.41860  +. 


101.  If  the  numerator  and  denominator  of  a  fraction  are  not  perfect 
squares,  how  do  you  extract  the  square  root  ? 


EXTRACTION  OF  THE  SQUARE  ROOT.    167 

13 

5.  What  is  the  square  root  of  7—  ?         Ans.  2.7131 3  + . 

36 

15 

6.  What  is  the  square  root  of  8—  1         Ans.  2.88203  -f . 

u  49 

5 

7.  What  is  the  square  root  of    —  1         Ans.  0.64549  +. 

3 

8.  What  is  the  square  root  of    10—  1 

Ans.  3.20936  +. 

102.  Finally,  instead  of  the  last  method,  we  may,  if  we 
please, 

Change  the  vulgar  fraction  into  a  decimal,  and  continue  the 
division  until  the  number  of  decimal  places  is  double  the  num- 
ber of  places  required  in  the  root.  Tlien,  extract  the  root  of 
the  decimal  by  the  last  rule. 

11 

Ex.  1.  Extract  the  square  root  of    —     to  within  .001. 

This  number,  reduced  to  decimals,  is  0.785714  to  within 
0.000001  ;  but  the  root  of  0.785714  to  the  nearest  unit,  is 

11 

.886:  hence  0.886  is  the  root  of    —    to  within  .001. 

14 

/      1  ° 

2.  Find  the  \/ 2-^   to  within  0.0001. 

v     15 

Ans.  1.6931  +. 

1 

3.  What  is  the  square  root  of    —  ?        Ans.  0.24253  +. 

7 

4.  What  is  the  square  root  of  —  ?         Ans.  0.93541  -f-. 

8 

5 

5.  What  is  the  square  root  of  —  ?         Ans.  1.29099  +. 


102.  By  what  other  method  may  the  root  be  found  f 


108  ELEMENTARY     ALGEBRA 


Extraction  of  the  Square  Root  of  Monomials. 

103.  In  order  to  discover  the  process  for  extracting  the 
square  root,  we  must  see  how  the  square  of  a  monomial  is 
formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  35), 
we  have 

(5a2Z>3c)2  =  5a2b3c  X  oa2b3c  =  2oaibsc2  ; 

that  is,  in  order  to  square  a  monomial,  it  is  necessary  to 
square  its  co-efficient,  and  double  the  exponents  of  each  of  the 
different  letters.  Hence,  to  find  the  square  root  of  a  mono- 
mial, we  have  the  following 

RULE. 

I.  Extract  the  square  root  of  the  co-efficient. 
II.  Divide  the  exj^onent  of  each  letter  by  2. 

Thus,      v/o4^6i  —  8a3b2    for     8a3b2  x  8a362  =  04a654. 

2.  Find  the  square  root  of  625a258c6.  Ans.  25oi4cn. 

3.  Find  the  square  root  of  570a4i6c8.  Ans.  24a263c4. 

4.  Find  the  square  root  of   \d&x&y2zi.  Ans.  l-ix3  t/z7. 

5.  Find  the  square  root  of  441a8£6c10c/16. 

Ans.  21aib3ccd8. 
0.  Find  the  square  root  of  784a12&wc16<22. 

Ans.  2Sa66W 
7.  Find  the  square  root  of  81a864c6. 

Ans.  Qa'tfc3, 


103.  How  do  you  extract  the  square  root  of  a  monomial  t 


EXTRACTION  OF  THE  SQUiRE  ROOT    1G9 

104.  From  the  preceding  rule  it  follows,  that  vhen  a 
monomial  is  a  perfect  square,  its  numerical  co-efficient  is  a 
perfect  square  and  all  its  exponents  even  numbers.  Thus, 
25«4&2  is  a  perfect  square ;  but  98u&4  is  not  a  perfect  square, 
because  08  is  not  a  perfect  square,  and  a  is  affected  with 
an  uneven  exponent. 

In  the  latter  case,  the  quantity  is  introduced  into  the  cal- 
culus by  affecting  it  with  the  sign  -y/  ,  and  it  is  written 
thus : 

Quantities  of  this  kind  are  called  radical  quantities,  t  irra- 
tional quantities,  or  simply  radicals  of  the  second  degree. 
They  are  also,  sometimes  called  Surds. 

Such  expressions  may  often  be  simplified,  by  employing 
the  principle  that,  the  square  root  of  the  product  of  two  or 
more  factors  is  equal  to  the  product  of  the  square  roo*",  of 
these  factors ;  or,  in  algebraic  language, 

■y/ubed  .  .  .   =y/'a.-y/&.y'c.  -\/  d  .  .  . 


This  being  the  case,  the  above  expression,    y98abl      :an 
be  put  under  the  form 


•v/496*  x2a=  y^9P  X'  </2a. 

Now,   -y/4964,  may  be  reduced  to  7£2;  hence, 


-y/98^*  =  lbz  V2«- 
In  like  manner, 

■y/iSaW^d  =  i/QaWc2  X  5bd  =  Sabc  ^/EbS. 


^/8C4a265cn  =  yT44u2W°x66c  =  12t/62c5  y/Gbc, 
H 


170  ELEMENTARY     ALGEBRA. 

The  quantity  which  stands  without  the  radical  sign  ia 
called  the  co-efficient  of  the  radical.  Thus,  in  the  expres- 
sions 

752y^a7    Sabc^/bbd,     I2ab2c5^/6bc, 

the  quantities  lb2,  oabc,  12ab2c5,  are  called  co-efficients  of  the 
radicals. 

Hence,  to  simplify  a  radical  expression  of  the  second 
degree,  we  have  the  following 

RULE. 

I.  Separate  the  expression  under  the  radical  sign  into  two 
factors,  one  of  which  shall  be  a  perfect  square. 

II.  Extract  the  square  root  of  the  perfect  square,  and  then 
multiply  the  root  by  the  indicated  square  root  of  the  remaining 
factors. 

105.  Remark. — To  determine  if  a  given  number  has  any 
factor  which  is  a  perfect  square,  we  examine  and  see  if  it  ia 
divisible  by  either  of  the  perfect  squares 

4,     9,     1G,     25,     36,     49,     04,     81,  &c, 

and  if  it  is  not,  we  conclude  that  it  does  not  contain  a  factor 
which  is  a  perfect  square. 


104.  When  is  a  monomial  a  perfect  square  ?  "When  it  is  not  a  perfect 
Bquare,  how  is  it  introduced  into  the  calculus  ?  What  are  quantities  of 
this  kind  called  ?  May  they  be  simplified  ?  Upon  what  principle  ? 
What  is  a  co-efficient  of  a  radical  ?  Give  the  rule  for  reducing  radi- 
cals. 

105.  How  do  you  determine  whether  a  given  number  has  a  factor 
which  is  a  perfect  square  ? 


EXTRACTION     OF     THE     SQUARE     ROOT.        171 


EXAMPLES. 


1.  Reduce     -^75a3bc     to  its  simplest  form. 


Ans.  5a  y/Sabc 


2.  Reduce     ^/128b5a6d2     to  its  simplest  form. 

Ans.  SPa3d^2b. 

3.  Reduce     ^/32a9b8c     to  its  simplest  form. 

Ans.  4a*64  -y/2ac. 

4.  Reduce     -y/256a264c8     to  its  simplest  form. 

Ans.  16a62c*. 


5.  Reduce     -y/1024a967c5     to  its  simplest  form. 

Am.  32a*b3c2^/abe. 

6.  Reduce     ^/129a!  b5c6d     to  its  simplest  form. 

Ans.  27a3b2c3^/abd. 

7.  Reduce     -y/675a7&5c26?     to  its  simplest  form. 

^bzs.  15a362c-/3^R 

8.  Reduce     y^l445a3c8(/4     to  its  simplest  form. 

Ans.  17ac4e?2-y/5a. 

9.  Reduce     -y/1008a9J7ra8     to  its  simplest  form. 

^ns.  12a4d3m*Wlad. 


10.  Reduce     -y/215Ga1068c6     to  its  simplest  form. 

Ans.  14a5Mc3^/lT. 

11.  Reduce     y'405a7i6c?8     to  its  simplest  form. 


172  ELEMENTARY      ALGEBRA. 

106.  Since  like  signs  in  two  factors  give  a  plus  sign  in 
the  product,  the  square  of  —  a,  as  well  as  that  of  -f-  a,  will 
be  a2 ;  hence,  the  square  root  of  a2  is  either  -\-  a  or  —  a. 
Also,  the  square  root  of  25«264  is  either  +  hab2  or  —  5a62, 
Whence  we  may  conclude,  that  if  a  monomial  is  positive, 
its  square  root  may  be  affected  either  with  the  sign  +  or  -  ; 
thus,  -y/tJa*  =  ±  3a2  ;  for,  +  3a2  or  —  3a2,  squared,  g.ves 
9a4.  The  double  sign  rir  with  which  the  root  is  affected,  is 
read  plus  or  minus. 

If  the  proposed  monomial  were  negative,  it  would  be  im- 
possible to  extract  its  square  root,  since  it  has  just  been 
shown  that  the  square  of  every  quantity,  whether  positive 
or  negative,  is  essentially  positive.     Therefore, 


-y^-  9,     -/-  4a2,     y— 8a26, 

are  algebraic  symbols  which  indicate  operations  that  cannot 
be  performed.  They  are  called  imaginary  quantities,  or 
rather,  imaginary  expressions,  and  are  frequently  met  with 
in  the  resolution  of  equations  of  the  second  degree.  These 
symbols  can,  however,  by  extending  the  rules,  be  simplified 
in  the  same  manner  as  those  irrational  expressions  which 
indicate  operations  that  cannot  be  exactly  performed.  Thus, 
■y/  —  9  may  be  reduced  by  (Art.  104).     Thus, 


V  -  9 

:  V 

'4aTX 

r=3  ^ 

=  2a 

2a 

-1, 
~\  : 

y/2b> 

and 

—  4a2  = 

also, 

V- 

-Sazb 

=  y/W 

X  - 

-26  = 

2a  y^ 

26  = 

<V-i. 

106.  What  sign  is  placed  before  the  square  root  of  a  monomial 
Why  may  you  place  the  sign  plus  or  minus  ?  What  is  an  imaginan 
quantity  ?     Why  is  it  called  imaginary  ? 


RADICALS  OF  THE  SECOND  DEGREE.    173 

Of  the  Calculus  of  Radicals  of  the  Second  Degree. 

107.  A  radical  quantity  is  the  indicated  root  of  an  im 
perfect  power. 

The  extraction  of  the  square  root  gives  rise  to  such  expres- 
sions as  i/a~,  oi/b,  7 -\/%  which  are  called  irrational 
quantities,  or  radicals  of  the  second  degree.  We  will  now 
establish  rules  for  performing  the  four  fundamental  opera- 
tions of  Algebra  upon  such  expressions. 

108.  Two  radicals  of  the  second  degree  are  similar,  when 
the  quantities  under  the  radical  sign  are  alike  in  both.    Thus, 

3-y/iT  and   bc^/b    are    similar   radicals;    and    so    also  are 

9yT  and   7-/2T 

Addition. 

109.  Radicals  of  the  second  degree  may  be  added  together 
by  the  following 

* 
RULE. 

I.  If  the  radicals  are  similar  add  their  co-efficients,  and  to 
the  sum  annex  the  common  radical. 

II.  If  the  radicals  are  not  similar,  connect  them  together 
with  their  proper  signs. 

Thus,  3a  yT+  5c  ^/T=  (3a  +  5c)  ^/b7 


107i  What  is  a  radical  quantity  ?     What  are  such  quantities  called  ! 
108.  When  are  radicals  of  the  second  degree  similar? 
10^a  How  do  you  ad</  similar  radicals  of  the  second  degree  ?     How 
do  you  add  radicals  which  are  not  similar  ? 


174  ELEMENTARY     ALGEBRA. 

In  like  manner, 

7  v/2a~+  3  V*1  =  (7  +  3)  V**  =  10  V**- 

Two  radicals,  which  do  not  appear  to  be  similar  at  first 
sight,  may  become  so  by  simplification  (Art.  104). 

For  example, 
-yASai2  +  6  -/JSa  =  46  -/3a  +  56  -/3a  =s  96  -/3a] 

and        2  v/45-f-3-v/5  =  Gv/5  +  3 -^=9  V^ 

When  the  radicals  are  not  similar,  the  addition  or  sub- 
ti  ictior:  can  only  be  indicated.  Thus,  in  order  to  add 
3  \/b~  to    5  -y/a^   we  write 

5  Va  +  3  \/bT 

EXAMPLES. 

1.  What  is  the  sum  of    ./27a2    and     -/48a2  ? 

-4  ns.  7a -/3T 

2.  What  is  the  sum  of    v^OoM2    and    «/72a*62  ? 

-4n*.  lla2fl-/5T 

/3^2~  /  a2 

3.What  is  the  sum  of  ^  -r —  and   \J  — r=-  * 


<4«s.  4a  \/ . 

V    15 

4.  What  is  the  sum  of    ./T25     and     -/500a2 1 

Ana.  (5  +  10a)  y^ 


RADICALS  OF  THE  SECOND  DEGREE.    175 


5.  What  is  the  sum  of  x  f3*L    and   ,  /—  ? 
V   147  V  294  ' 


A  10        r— 


6.  What  is  the  sum  of    ^/dSa^x    and  -^/oQx2  —  86«2  ? 


-4n«.  70^/2*  +  6-y/x2  -  a2. 


7.  What  is  the  sum  of    -/98o2*    and  y^SSu4*5  ? 

-4n*.  (7a  +  12a2x2)  y/^ 

8.  Eequired  the  sum  of  -y/72    and  y/128. 

9.  Eequired  the  sum  of    -y/27     and  y/l47. 

Jns.  10-v/aT 

/2  /27 

10.  Eequired  the  sum  of    ♦  / —  and  a  / . 

^  V  3  V  50 

11.  Eequired  the  sum  of    2y/o2&     and     3-y/64ix4. 

Ans.  (2a  +  24x2)y^r 

12.  Eequired  the  sum  of    -y/243     and     10y/3G3. 

^tw.  119  ^/a. 

13.  What  is  the  sum  of    y/S20a2b2     and  v/245a8i6  ? 

.4m?.  (8a&  +  7a4&3)  y^ 

14.  What  is  the  sum  of    y/  75a667    and    y/300a665  ? 

^ne.  (5a3&3  +  lOa^y/sZ 


17G  ELEMENT  AKY     ALGEBRA. 

Subtraction. 

110.  To   subtract  one  radical  expression  from  another, 
we  have  the  following 

RULE. 

I.  If the  radicals  are  similar,  subtract  their  co-efficients,  and 
to  the  difference  annex  the  common  radical. 

II.  If  the  radicals  are  not  similar,  indicate  their  difference 
by  the  minus  sign. 

EXAMPLES. 

1.  What  is  the  difference  between    3a  yb    and    aybl 
Here,     3a  t/T  —  a  yfb  =  2a  yb     Ans. 

2.  From     9a  y/VW-   subtract   6a  ^/'Zlb2. 

First,     9a  v^?l2  =  27 'ah  y^  and   6a  ^U2  =  18ab  /IT; 
and  27abi/~3—  18a&*/~3  =  9a&-/~3    Ans. 

3.  What  is  the  difference  of  -^75    and  -y/48  1 

Ans.  -y/Z. 


4.  What  is  the  difference  of  v/24a262    and    -/546«  ? 

^4/is.   (2a6-362)v/0T 


110.   How  do  you  subtract  similar  radicals?     How  do  ycu  suhtiacx 
radicals  which  arc  not  oiuiilar  ? 


RADICALS   OF  THE  SECOND  DEGREE.    177 


5.  Required  the  difference  of  \  /— -    and    \/— 

V  5  V  27 


Ans.  — 


45 


=  V1{ 


G    What  is  the  difference  of     yT28a3P    and     -/Saa9  ] 

Ans.  (Sab  —  4a1) -/2a 

7.  What  is  the  difference  of     -^/iSa^o3    and     y/[)ub  1 

Ans.  4.ab -y/'iub  —  3  ^/ab 

8.  What  is  the  difference  of   -v/242a5F    and     ^/2a3b3  % 

Ans.  (1  la2b2  —  ab)  -y/2a6 


9.  What  is  the  difference  of  \/—     and  \/  —  ? 

4  V   9 


./ins VIT 

G 


10.  What  is  the  difference  of  -/320a2    and    -v/SOo2? 

Ans.  4a-y/5. 

11.  What  is  the  difference  between 


-v/720a:i63     and     y245aic2cZ2  ? 

^4«s.  (12ai  —  7ccZ)  -v/506. 
12    What  is  the  difference  between 


ylJGS^F     and     y/200a2F  ? 


^ns.  1205^. 


18    What  is  the  difference  between 


VTl2a866     and     -v/28a866  ? 


8* 


Ans.  2a^V7 


178  ELEMENTARY    ALGEBRA. 


Multiplication. 

111.    For  the   multiplication  of  radicals,  we  have  the 
following 

RULE. 

I.  Multiply  the  quantities  under  the  radical  signs  together, 
and  place  the  common  radical  sign  over  the  product. 

II.  If  the  radicals  have  co-efficients,  we  multiply  them  to- 
gether, and  place  the  product  before  the  radical  part. 


Thus, 


-vATx  i/~b  =  i/ah; 


This  is   the  principle  of  Art.  104,  taken  in  the  inverse 
order. 


EXAMPLES. 


1.  What  is  the  product  of   3   i/bab    and    4v/20a? 

Ans.  120a  -/X 

2.  What  is  the  product  of    2a  \/bc     and    3a  ^/Tc  ? 

Ans.   Qa2bc. 

3.  What  is  the  product  of  2a^/a2  +  b2  and   —3a  ya2  +  b21 

Ans.   —  Qa2  (a2  +  b2). 


1 1 1  •  How  do  you  multiply  quantities  which  are  under  radical  signs  f 
When  the  radicals  have  co-efficients,  how  do  you  multiply  them 


RADICALS  OF  THE  SECOND  DEGREE.   179 

4.  What  is  the  product  of   3  </T  and  2yr8  1 

Ans.  24. 

5.  What  is  the  product  of   fyl"26    and     i2o  VI ^  • 

6.  What  is  the  product  of  2x  +  V^  an<l    2a;  —  -y/b  1 

Ans.  4x2  —  b. 

7.  What  is  the  product  of 


Va-i-2^/T    and     V a  —  %y/b  % 

Ans.  -\/a?~—  4£>. 


8.  What  is  the  product  of   3a  y^  27a3    by     y2o? 

Ans.  9a3 -/6. 

Division, 

112.  To  divide  one  radical  by  another,  we  have  the  fol- 
lowing 

RULE. 

I.  Divide  one  of  the  quantities  under  the  radical  sign  by 
the  other,  and  place  the  common  radical  sign  over  the  quotient. 

II.  If  the  radicals  have  co-efficients,  divide  the  co-efficient  of 
the  dividend  by  the  co-efficient  of  the  divisor,  and  place  the 
quotient  before  the  radical,  found  as  above. 


112.   now  do  you  divide  quantities  ■which  are  under  the  radical  sign  1 
When  the  radicals  have  co-efficients,  how  do  you  divide  them  ? 


180 


ELEMENTARY     ALGEBEA. 


Thus,     ^L___*/ — ;     for   the    squares   of    these    two 
Vb        V    b 

a 
expressions    are    each    equal    to    the    same  quantity     — ; 

hence  the  expressions  themselves  must  he  equal. 


EXAMPLES. 


1.  Divide     5a  -/b     by     26  -/c. 

12ac  -/Obc  by  4c  -/2b. 
Q»a  ^966*  by  3  t/W. 
4a2y/5065     by     2a2y/57. 


2.  Divide 

3.  Divide 

4.  Divide 

5.  Divide 


5a    fb 

Ans.  7rr\/  — . 
26  V    c 

Ans.  3a  y3c. 

Ans.  4ab  i/~3. 

Ans.  262iA0^ 


2Ga36y/81a262     by     13a-/9a£ 

Ans.  Qa2b/ab. 
6.  Divide     84a364 y7 27ac     by     42a5y/3a^ 

Ga^y7  20a3     by     12-/5a^ 


7.  Divide 

8.  Divide 

9.  Divide 

10.  Divide 

11.  Divide 


Ga-y/1062     by     3-/5T 
4864y/T5     by     262y/7J. 
SerWy7?^     by     20^/28^ 


Ans.  a3b2. 
Ans.  2o6y/2. 
^4ns.  3G062. 


Ans.  2ab*c3d. 


12.  Divide     96aic3-/9SbE     by     48a6c-y/2T. 


.<4ns.  14a36r2. 


RADICALS     OF     THE     SECOND     DEGREE.         181 


13.  Divide     27a6b9</2La?     by     ./7a. 

Am.  viaWy/'W. 

14.  Divide     18a848-/8a*     by     6aiy^; 

-4>w.  6a865-/2~ 

■    7b  Extract  the  Square  Root  of  a  Polynomial. 

113.  Before  explaining  the  rule  for  the  extraction  of 
tin  square  root  of  a  polynomial,  let  us  first  examine  the 
squares  of  several  polynomials  :  we  have 

(a  +  b)2  =  a2  +  2ab  +  b2, 

(a  +  b  -f  c)2  =  a2  +  2ab  +  b2  +  2(a  +  4)c  -f-  c2, 

(a  +  5  +  c  +  a7)2  =  a2  +  2a£  +  62  -f  2(a  +  <j)c  +  c8 

+  2(a  +  6  +  c)d  -f  <22. 

The  law  by  which  these  squares  are  formed  can  be  enun- 
ciated thus : 

The  square  of  any  polynomial  is  equal  to  the  square  of  the 
first  term, plus  twice  the  product  of  the  first  term  by  the  second, 
plus  the  square  of  the  second ;  plus  twice  the  first  two  ter?ns 
multiplied  by  the  third,  plus  the  square  of  the  third  ;  plus  twice 
the  first  three  terms  multiplied  by  the  fourth,  plus  the  square 
of  the  fourth  ;  and  so  on. 


113.  What  is  the  square  of  a  binomial  equal  to  What  is  the 
aquare  of  a  trinomial  equal  to  ?  What  is  the  square  of  any  polynomial 
equal  to  f 


182  ELEMENTARY     ALGEBRA. 

114.  Hence,  to  extract  the  square  root  of  a  polynomial, 
we  have  the  following 

RULE. 

I.  Arrange  the  polynomial  with  reference  to  one  of  its  let- 
ters, and  extract  the  square  root  of  the  first  term:  this  will 
aire  the  first  term  of  the  root. 

II.  Divide  the  second  term  of  the  polynomial  by  double  the 
first  term  of  the  root,  and  the  quotient  will  be  the  second  term 
of  the  root. 

III.  Then  form  the  square  of  the  sum  of  the  two  terms  of 
the  root  found,  and  subtract  it  from  tl/e  first  polynomial,  and 
then  divide  the  first  term  of  the  remainder  by  double  the  first 
term  of  the  root,  and  the  quotient  will  be  the  third  term. 

IV.  Form  the  double  product  of  the  sum  of  the  first  and 
second  terms  by  the  third,  and  add  the  square  of  the  third ; 
then  subtract  this  result  from  the  last  remainder,  and  divide 
the  first  term  of  the  result  so  obtained  by  double  the  first  term 
of  the  root,  and  the  quotient  will  be  the  fourth  term.  Then 
proceed  in  a  similar  manner  to  find  the  other  terms. 

EXAMPLES. 

1.  Extract  the  square  root  of  the  polynomial 

49a262  —  24ab3  -f  25a4  -  30a3b  +  16bK 
First  arrange  it  with  reference  to  the  letter  a. 


25a*  —  SOa^b  +  49a2/;2  —  2iaP  —  IC64 
25a*  —  S0a3b  +    9aW 


5a2  —  3ah  4  45s 
10a2~ 


40a2&2  -  24a&3  +  IGo*    1st  Kern. 
40a262  —  24a63  +  lQb* 

0     .     .     .      2d  Rem. 


RADICALS  OF  THE  SECOND  DEGREE.    183 

After  having  arranged  the  polynomial  with  reference  to 
a,  extract  the  square  rout  of  25a4 ;  this  gives  5a2,  which  is 
placed  at  the  right  of  the  polynomial :  then  divide  the  second 
term,  —  SQa3b,  by  the  double  of  5a2,  or  10a2  ;  the  quo- 
tient is  —  Sab,  which  is  placed  at  the  right  of  5a2.  Hence, 
the  first  two  terms  of  the  root  are  5a2  —  oab.  Squaring 
this  binomial,  it  becomes  25a4  —  30a36  +  9a262,  which,  sub- 
tracted from  the  proposed  polynomial,  gives  a  remainder, 
of  which  the  first  term  is  40u2b2.  Dividing  this  first  term 
by  10a2,  (the  double  of  5a2),  the  quotient  is  +  462 ;  this 
is  the  third  term  of  the  root,  and  is  written  on  the  right  of 
the  first  two  terms.  By  forming  the  double  product  of 
5a2  —  oab  by  4b2,  squaring  4b2,  and  taking  the  sum,  we 
find  the  polynomial  40a262  —  24ab3  -f  1064,  which,  sub 
tracted  from  the  first  remainder,  gives  0.  Therefore, 
5a2  —  oab  +  462  is  the  required  root. 

2.  Find  the  square  root  of  a4  +  Aa3x  4-  6a2x2  4-  4a23  4-  *4. 

Ans.  a2  4-  2«x  4-  z2. 

3.  Find  the  square  root  of  a4  —  4a3.z  +  Ga2*'2  —  4«.r3  +  X*. 

Ans.   a2  —  2az  -\-  x2. 

4.  Find  the  square  root  of 

4z6  4-  12.c5  4-  5x*  —  2x3  4-  7a:2  —  2x  4-  1. 

Ans.   2x3  4-  3x2  -  x  4-  1. 

5.  Find  the  square  root  of 

9a4  -  \2a3b  4-  28a2/,2  -  16ai3  +  Wb\ 

Ans.    3a2  —  2a6  4- 462 


114.  Give  the  rule  for  extracting  the  square  root  of  a  polynomial 
What  is  the  first  step  ?  "What  the  second  ?  What  the  third  J  WhaJ 
the  fourth? 


184  ELEMENTARY      ALGEBKA. 

6.  What  is  the  square  root  of 

xi  —  lax3  +  4a2x2  —  4t2  +  Sax  4-  4  1 

Ans.  z-  —  2ax  —  & 

7.  What  is  the  square  root  of 

9x2  —  12.r  +  Qxy  4-  y2  —  4y  +  4 1 

^4»s.  3z  4-  y  —  2. 

8.  What  is  the  square  root  of   y4  —  2y2x2  4-2.e2  +  2y2 

4- 1  -f.  jc*  ?  ^4»5.  r  —  x2  —  i. 

9.  What  is  the  square  root  of    SaW  —  30a363  4-  25a262  ? 

Ans.  oa2b2  —  5ab. 

10.  Find  the  square  root  of 

25a462  —  40a3b2c  4-  7Qa2b2c2  —  4Sa52c3  +  3GJ2c4  —  30a4Sc 
+  24a36c2  —  3Ga25c3  4-  9a4c2. 

Ans.    5a2b  —  3a2c  —  4abc  4-  Qbc2. 

115.  We  will  conclude  this  subject  with  the  following 
remarks  : 

1st.  A  binomial  can  never  be  a  perfect  square,  since  we 
know  that  the  square  of  the  most  simple  polynomial,  viz : 
a  binomial,  contains  three  distinct  parts,  which  cannot  ex- 
perience any  reduction  amongst  themselves.  Thus,  the 
expression  a2  +  b2  is  not  a  perfect  square ;  it  wants  the 
term   ±2ab  in  order  that  it  should  be  the  square  of  a  ±  b. 

2d.  In  order  that  a  trinomial,  when  arranged,  may  be  a 
perfect  square,  its  two  extreme  terms  must  be  squares,  and 
the  middle  term  must  be  the  double  product  of  the  square 
roots  of  the  two  others.  Therefore,  to  obtain  the  square 
root  of  a  trinomial  when  it  is  a  perfect  square  :  Extract  the 
roots  of  the  two  extreme  terms,  and  give  these  roots  the  same 
or  contrary  signs,  according  as  the  middle  term  is  positive  or 


RADICALS     OF     THE     SECOND     DEGREE,  18fj 

negative.      To  verify  it,  see  if  the  double  product  of  the  two 
roots  is  the  same  as  the  middle  term  of  the  trinomial.     Thus, 

9a6  —  48a462  +  64a26*   is  a  perfect  square, 


since  y9uP  =  3a3,    and  y/C4a264  =  —  8ab2, 

and  also  2  x  3a3  x  —  Sab2  =z  —  48a462  =  the  middle  term. 

But,  4a2  +  14a6  +  962  is  not  a  perfect  square  :  for, 
although  4a2  and  +  9b2  are  the  squares  of  2a  and  36, 
yet    2  X  2a  X  36   is  not  equal  to    14tt&. 

3d.  In  the  series  of  operations  required  by  the  general 
rule,  when  the  first  -term  of  one  of  the  remainders  is  not 
exactly  divisible  by  twice  the  first  term  of  the  root,  we  may 
conclude  that  the  proposed  polynomial  is  not  a  perfect 
square.  This  is  an  evident  consequence  of  the  course  of 
reasoning,  by  which  we  have  arrived  at  the  general  rule  for 
extracting  the  square  root. 

4th.  When  the  polynomial  is  not  a  perfect  square^  it  may 
sometimes  be  simplified.  (See  Art.  104.) 


Take,  for  example,  the  expression  -y/a36  +  4alb2  +  4a63. 

The  quantity  under  the  radical  is  not  a  perfect  square ; 
but  it  can  be  put  under  the  form  ab  [a,2  +  4cr6  4-  462.) 
Now,  the  factor  within  the  parenthesis  is  evidently  the 
square  of  a  +  26,  whence  we  may  conclude  that, 


■yfo?b  +  4a262  +  4a63  =  (a  4-  26)  </ab. 


2.  Reduce  y2a?b  —  4ab2  +  263   to  its  simple  form. 

Am.  (a  —  6)  V2X 

115.  Can  a  binomial  ever  be  a  perfect  power?    Why  not  ?     When  is 
a  trinomial  a  perfect  square  ?     When,  in  extracting  the  square  root,  we 
find  that  the  first  term  of  the  remainder  is  not  divisible  by  twice  the 
toot,  is  the  polynomial  a  perfect  power  or  not  ? 
9 


!8fi  ELEMENTARY    ALGEBRA. 


CHAPTER  VI. 

Equations  of  the  Second  Degree. 

116.  An  Equation  of  the  second  degree  is  one  in  -which 
the  greatest  exponent  of  the  unknown  quantity  is  equal  to  2 

If  the  equation  contains  two  unknown  quantities,  it  is  of 
the  second  degree  when  the  greatest  sum  of  the  exponents 
with  which  the  unknown  quantities  are  affected,  in  any 
term,  is  equal  to  2.     Thus, 

x1  =.  o,     ax1  +  hx  —  c,     and    xy  -f-  x  =  d2, 

are  equations  of  the  second  degree. 

117.  Equations  of  the  second  degree,  involving  a  single 
unknown  quantity,  are  divided  into  two  classes : 

1st.  Equations  which  involve  only  the  square  of  the  un- 
known quantity  and  known  terms.  These  are  called  Incom- 
plete Equations. 

2d.  Equations  which  involve  the  first  and  second  powers 
of  the  unknown  quantity  and  known  terms.  These  are 
called  Complete  Equations.  • 


lid  What  is  an  equation  of  the  second  degree  f 
117i  Into  how  many  classes  are  equations  of  the  second  degree  di- 
vided ?    What  is  an  incomplete  equation  !  "What  is  a  complete  equation  f 


EQUATIONS  OF  THE  SECOND  DEGREE.   1P7 

Thus,  x2  +  2x2  -  5  =  7 

and  5x2  —  3x2  —  4  =  a 

are  incomplete  equations  :  and 

$x2  —  5x   —  3x2  +  a  =  b 

2x:'  —  8x2—    x  —  c  =  d 

are  complete  equations. 

Of  Incomplete  Equations. 

118.  If  we  take  an  incomplete  equation  of  the  form 

14a;2  —  Sx2  =  40  —  2x2 

we  have,  by  collecting  the  terms  involving  x2, 

8x2  =  40,  or  x2  =  5. 

Again,  if  we   have  the  equation 

ax2  +  bx2  +  d=f 

we  shall  have, 

f j 

(a  +  b)x2  =  f—  d,  and  x2  = -  =  m. 

a  +  o 

by  substituting  m  for  the  known  terms  which  compose  the 
second  member.     Hence, 

Every  incomplete  equation  can  be  reduced  to  an  equation 
involvi7ig  two  terms,  of  the  form 

x2  =  m, 
and  from  this  circumstance  the  incomplete  equations  aro 
often  called  equations  involving  two  terms. 

From  which  we  have,  by  extracting  the  square  root  of 

both  members,  ,      , — 

x  =  dc  y  in. 

1 18*  To  what  form  may  every  incomplete  equation  be  reduced  }    What 
are  incomplete  equations  often  called  t 


188  ELEMENTARY     ALGEBEA. 

1.  What  number  is  that  which  being  multiplied  by  itself 
the  product  will  be  144 1 

Let  x  =  the  number :  then 

x  X  x  =  x2  =.  144. 

It  is  plain  that  the  value  of  x  will  be  found  by  extracting 
the  square  root  of  both  members  of  the  equation :  that  is 

i/H?=  -yj  144  :  that  is,  x  =  12. 

2.  A  person  being  asked  how  much  money  he  had,  said 
if  the  number  of  dollars  be  squared  and  6  be  added,  the 
sum  will  be  42 :  how  much  had  he  ] 

Let  x  =  the  number  of  dollars. 

Then,  by  the  conditions 

x2  +  6  =  42  : 

hence,  x2  —  42  —  6  =  36. 

and  x  =  6. 

Ans.  $6. 

3.  A  grocer  being  asked  how  much  sugar  he  had  sold  to  a 
person,  answered,  if  the  square  of  the  number  of  pounds  be 
multiplied  by  7,  the  product  will  be  1575.  How  many 
pounds  had  he  sold? 

Denote  the  number  of  pounds  by  x* 

Then  by  the  conditions  of  the  question 

7x2  =  1575 : 

hence,  x2  =  225 

and  x  =  15. 

Ans.  15. 


EQUATIONS  OF  THE  SECOND  DEGREE.   189 

4.  A  person  being  asked  his  age  said,  if  from  the  square 
of  my  age  you  take  192,  the  remainder  will  be  the  square 
of  half  my  age :  what  was  his  age  ? 

Denote  his  age  by  x. 

Then  by  the  conditions  of  the  question 


193=(H'=T' 


and  by  clearing  the  fractions 

4x2  —  768  =  x2 ; 
hence,  4a;2  —  x2  =  7G8 

and  3x2  =  7G8 

x2  =  256 
x  -    16. 


Ans.  1G 


5.  What  number  is  that  whose  eighth  part  multiplied  by 
its  fifth  part  and  the  product  divided  by  4,  shall  give  a  quo- 
tient equal  to  40  ? 

Let  x  =  the  number. 

By  the  conditions  of  the  question 


(_L*XI*).,4  =  40, 


To0  =  4° 

by  clearing  of  fractions, 

x2  =  6400 
x   =    80. 


Ans.  80. 


190  ELEMENTARY     ALGEBRA. 

119.  Hence,  to  find  the  value  of  x  we  have  the  follow 
tog 

RULE. 

I.  Find  the  value  of  x2  ;  and  then  extract  the  square  root 
of  both  members  of  the  equation. 

6,  What  is  the  value  of  x  in  the  equation 

3z2-f  8  =  5z2  — 10. 

By  transposition  ox2  —  5x2  =  —  10  —  8, 

by  reducing  —  2a;2  =  —  18, 

by  dividing  by  2,  and  changing  the  signs 

z2  =  9, 

by  extracting  the  square  root,  x  =  3. 

We  should,  however,  remark  that  the  square  root  of  9,  is 
either     -f-  3    or    —  3.     For, 

+  3  X  +  3  =  9     and     —  3  x  —  3  =  9. 

Hence,  when  we  have  the  equation 

x2  =  9, 
we  have,  x  =  -j-  3     and     x  —  -—  3. 

120.  A  root  of  an  equation  is  any  expression  which  being 
substituted  for  the  unknown  quantity,  will  satisfy  the  equa- 
tion, that  is,  render  the  two  members  equal  to  each  other. 
This,  in  "the  equation 

*2  =  9 

there  are  two  roots,   +  3  and  —  3  ;    for  either  of  these 
numbers  being  substituted  for  x  will  satisfy  the  equation. 


EQUATIONS     OF     THE     SECOND     DEGREE.       191 

7.  Again,  if  we  take  the  equation 

x2  =  m, 
we  shall  have 

sc  =  +  -yfm,  and    x  =  —  <yfm. 
For,  (+</m)2=:m; 

and  (  —  -yj~m  )  2  =  m  ; 

Hence,  we  may  conclude, 

1st.  27ta^  every  incomplete  equation  of  the  second  degree 
has  two  roots. 

2d.  That  these  roots  are  numerically  equal,  but  have  con- 
trary signs. 

8.  "What  are  the  roots  of  the  equation 

3j2  +  G  =  4x2  —  10. 

Ans.  x  =  +  4  and  x  =  —  4. 

0.  What  are  the  roots  of  the  equation 
1  x2 

—  Z2-8=:  —  +  10. 
o  1) 

-4rcs.  a;  =  +  0  and  x  =  —  9. 
10.  What  are  the  roots  of  the  equation 
4.c2  +  13  -  2x2  =  45. 

Ans.  x  =  +  4  and  x  =  —  4. 

119.  How  do  you  resolve  an  incomplete  equation? 

120i  What  is  the  root  of  an  equation?  What  are  the  roots  of  thp 
equation  x*  =  9  !  Of  the  equation  xl  =  wi?  How  many  roots  has  every 
incomplete  equation  ?     How  do  those  roots  ccimpare  with  each  other  I 


192  ELEMENTARY     ALGEBRA. 

8.  What  are  the  roots  of  the  equation 

Qx2  -  7  =  Sx2  +  5. 

Ans.  x  =  +  2,     x  =  -/-  2 

9.  "What  are  the  roots  of  the  equation 

re2 

8  +  5a;2  =   —  +  4x2  +  28. 

5 

Ans.  x  =  +  5,     a?  =  —  5. 

10.  Find  a  number  such  that  one-third  of  it  multiplied 
by  one-fourth  shall  be  equal  to  108 1 

Ans.  30. 

11.  What  number  is  that  whose  sixth  part  multiplied  by 
its  fifth  part  and  product  divided  by  ten,  shall  give  a  quo- 
tient equal  to  3  ? 

Ans.  30. 

12.  What  number  is  that  whose  square,  plus  18,  shall  be 
equal  to  half  its  square  plus  30  J 1 

Ans.  5. 

13.  What  numbers  are  those  which  are  to  each  other  as 
1  to  2  and  the  difference  of  whose  squares  is  equal  to  75  ? 

Let        x  =      the  less  number. 
Then  2x  =     the  greater. 
Then,  by  the  conditions  of  the  question 
4x2  —  x2  =  75, 
hence,  3.r2  =  75  ; 

and  by  dividing  by    3,   x2  =  25    and   2  =  5, 
and  2x  =  10. 

Ans.  5  and  10. 


EQUATIONS  OF  THE  SECOND  DEGREE.    193 

14.  What  two  numbers  are  those  which  are  to  each  other 
as  5  to  6,  and  the  difference  of  whose  squares  is  44  ? 

Let         x  =.    the  greatest  number. 

5 

Then,  — -  x  =  the  less. 

By  the  conditions  of  the  problem 

25 

x2  —  ^— x2  =  44 ; 
36 

by  clearing  of  fractions, 

SQx2  —  25x2  =  1584  ; 
hence,  11a;2  =  1584, 

and  x2  =  144, 

hence,  x  =  12, 

5 

and  ~x  =  10. 

o 

Ans.  10  and  12. 

15.  What  two  numbers  are  those  which  are  to  each 
other  as  3  to  4,  and  the  difference  of  whose  squares  is  28  ? 

Ans.  6  and  8 

16.  What  two  numbers  are  those  which  are  to  each  othej 
as  5  to  11,  and  the  sum  of  whose  squares  is  584  1 

Ans.  10  and  22. 

17.  A  says  to  B,  my  son's  age  is  one  quarter  of  your?, 
and  the  difference  between  the  squares  of  the  number? 
representing  their  ages  is  240  :  what  were  their  ages  1 

Eldest       16 


Ans.  . 

Younger     4 
9 


194  ELEMENTARY     ALGEBRA. 

Whin  tlsre  are  two  unknown  quantities. 

121.  When  there  are  two  or  more  unknown  quantities, 
eliminate  one  of  them  by  the  rule  of  Article  77:  there  ivill 
tints  arise  a  new  equation  with  but  a  single  unknown  quantity, 
the  value  of  which  may  be  found  by  the  rule  already  given. 

\.  There  is  a  room  of  such  dimensions,  that  the  differ- 
ence of  the  sides  multiplied  by  the  less,  is  equal  to  36,  and 
the  product  of  the  sides  is  equal  to  300  :  what  are  the 
sides  1 

Let  x  =  the  less  side ; 

y  =  the  greater. 
Then,  by  the  first  condition, 

(y-x)x  =  36; 
and  by  the  2d,  xy  =  360. 

From  the  first  equation,  we  have 

xy  —  x2  =  36  ; 
and  by  subtraction,  x2  =  324. 

Hence,  x  =  -^/324  =  18  ; 


360 
3/  =  ^  =  20. 


Ans.  x  =  l8,y  =  20. 


121  •  How  do  you  resolve  the  equation  'when  there  are  two  or  more 
unknown  quantities  ? 


EQUATIONS  OF  THE  SECOND  DEGREE.   195 

2.  A  merchant  sells  two  pieces  of  muslin,  which  together 
measure  12  yards.  lie  received  for  each  piece  just  so 
many  dollars  per  yard  as  the  piece  contained  yards.  Now, 
he  gets  four  times  as  much  for  one  piece  as  for  the  other : 
how  many  yards  in  each  piece  ? 

Let    x  =     the  number  in  the  larger  piece ; 
y  —    the  number  in  the  shorter  piece. 
Then,  by  the  conditions  of  the  question, 
x  +  y  =  12. 
x  x  x  =  x2  =  what  he  got  for  the  larger  piece  ; 
y  x  y  =y2  =  what  he  got  for  the  shorter. 
And  x2  =  4y2,  by  the  2d  condition, 

x  =  2y,  by  extracting  the  square  root. 

Substituting  this  value  of  x   in  the  first  equation,  we  have 
y  +  2y  =  12 ; 
and  consequently,  y  =   4, 

and  x  =    8. 

Ans.  8  and  4. 

3.  What  two  numbers  are  those  whose  product  is  30,  and 
quotient  3£  1  Ans.  10  and  3. 

4.  The  product  of  two  numbers  is  a,  and  their  quotient 
6  :  what  are  the  numbers  ? 


Ans.    yah  and  \/-r 


b' 

5.  The  sum  of  the  squares  of  two  numbers  is  117.  and 
the  diflerence  of  their  squares  45  :  what  are  the  numbers  ? 

Ans.  9  and  0. 


196  ELEMENTARY      ALGEBRA. 

6.  The  sum  of  the  squares  of  two  numbers  is  a,  and  the 
difference  of  their  squares  is  b  :  what  are  the  numbers  ? 


Ans.  x  =  yj  — — ,  y  =  ^  — - 


7.  What  two  numbers  are  those  which  are  to  each  other 
as  3  to  4,  and  the  sum  of  whose  squares  is  225  1 

Ans.  9  and  12. 

8.  What  two  numbers  are  those  which  are  to  each  othej 

as  m  to  n,  and  the  sum  of  whose  squares  is  equal  to  a2 1 

.  ma  na 

Ans.  , 

y  m2  -f-  n2       y»t2  -j-  «2 

9.  What  two  nuirAers  are  those  wThich  are  to  each  other 
as  1  to  2,  and  the  difference  of  whose  squares  is  75  ? 

Ans.  5  and  10. 

10.  What  two  numbers  are  those  which  are  to  each  othe; 
as  m  to  n,  and  the  difference  of  whose  squares  is  equa-1 
to  h2  % 

.  mb  lib 

Ans. 

y  m2  —  n2 '      y  m2  —  n2 

11.  A  certain  sum  of  money  is  placed  at  interest  for  siy 
months,  at  8  per  cent,  per  annum.  Now,  if  the  amount  bt 
multiplied  by  the  number  expressing  the  interest,  the  pro 
duct  will  be  $502500  :  what  is  the  amount  at  interest '? 

Ans.  $3750 

12.  A  person  distributes  a  sum  of  money  between  a  num 
ber  of  women  and  boys.  The  number  of  women  is  to  the 
number  of  boys  as  3  to  4.  Now,  the  boys  receive  one 
half  as  many  dollars  as  there  are  persons,  and  the  women 
twice  as  many  dollars  as  there  are  boys,  and  together  they 
receive. 138  dollars  :  how  many  women  were  there,  and  how 
many  hoys? 

36  women 
48  boys. 


Ans.  < 


EQUATIONS  OF  THE  SECOND  DEGREE.   1J)7 

Of  Complete  Equations. 

122.  We  have  already  seen  (Art.  117),  that  a  complete 
equation  of  the  second  degree,  contains  the  square  of  the 
unknown  quantity,  the  first  power  of  the  unknown  qvwiity.. 
and  known  terms. 

1.  If  we  have  the  complete  equation 

5x2  —  2x2  +  8  =  9x  +  32, 
we  have,  by  transposing  and  reducing, 
3x2  —  9x  =  24, 
and  by  dividing  by  3, 

■x2  —  Sx  =  8, 
an  equation  containing  but  three  terms. 

2.  If  we  have  the  equation 

a2x2  +  3abx  +  x2  =  ex  4-  d, 

by  collecting  the  co-efficients  of  x2  and  x,  we  lw  ■ 

(a2  +  l)x2  +  (Sab  —  c)x  =  d; 

and  dividing  by  the  co-efficient  of  x2,  we  have 

_       Sab  —  c  d 

x2  A x  = . 

*  T   a2  +  1  a2  +  1 


122.  Hew  many  terms  does  a  complete  equation  of  the  secend  degree 
contain  ?  Of  what  ia  the  first  term  compesed  ?  The  second  f  The 
third ! 


198  ELEMENTARY      ALGEBRA. 

If  wo  represent  the  co-efficient  of  x  by  2p,  and  the  known 
term  by  q.  we  have 

x2  -+-  2px  =  q, 
an  equation  containing  but  three  terms :    hence, 

Every  complete  equation  of  the  second  degree  may  be  re- 
duced to  an  equation  containing  but  three  terms. 

123.  We  wish  now  to  show  that  there  maybe  four  forms 
under  which  this  equation  will  be  expressed,  each  depending 
on  the  signs  of  2p  and  q. 

1st.  Let  us,  for  the  sake  of  illustration,  make 
2p  =  +  4,     and     q  =  -f  5 : 
we  shall  then  have         x2  -f  4a;  =  5. 

2d.  Let  us  now  suppose 

2p  =  —  4,     and     q  =  -f-  5  : 
we  shall  then  have         x2  —  A.x  =  5. 

3d.  If  wre  make 

2p  =  +  4,     and    g  =  —  5, 
we  have  x2  +  4z  =  —  5. 

4th.  If  we  make 

2p  =  —  4,     and    <?  =  —  5, 
we  have  x2  —  4r  =  —  5. 

123i  Under  how  many  forms  may  every  equation  of  the  second  de- 
gree be  expressed  ?  On  what  will  these  forms  depend  ?  What  are  the 
signa  of  the  co-efficient  of  x  and  the  known  term,  in  the  first  form  ? 
What  in  the  second?  Wha  in  the  third  ?  What  in  the' fourth?  Repeat 
the  four  forms. 


EQUATIONS  OF  THE  SECOND  DEGREE.    199 

Wc  therefore  conclude,  that  every  complete  equation  of 
the  second  degree  may  be  reduced  to  one  of  these  forms: 

x2  +  2px  =  -(-  q,  1st  form. 

x2  —  2/>x  =  -f-  q,  2d  form. 

a-2  -f-  %pz  =  —  1,  3d  form. 

x2  —  2px  =  —  q,  4th  form. 

124.  Remark. — If,  in  reducing  an  equation  to  either  of 
these  forms,  the  second  power  of  the  unknown  quantity 
shuuld  have  a  negative  sign,  it  must  be  rendered  positive 
by  changing  the  sign  of  every  term  of  the  equation. 

125.  We  are  next  to  show  the  manner  in  which  the  value 
of  the  unknown  quantity  may  be  found.  We  have  seen 
(Art.  38),  that 

{x  -\-p)2  =  x2  +  2px  +  p2 ; 

and  comparing  this  square  with  the  first  and  third  forms,  we 
see  that  the  first  member  in  each  contains  two  terms  of  the 
square  of  a  binomial,  viz  :  the  square  of  the  first  term  plus 
twice  the  product  of  the  2d  term  by  the  first.  If,  then,  wre 
take  half  the  co-efficient  of  x,  viz  :  p,  and  square  it,  and  add 
the  result  in  each  equation,  to  both  members,  we  have 

x2  -f-  2px  +  p2  =  q  +  p2, 
x2  -f-  2px  -f-  p2  =  —  q  -j-  2>2, 

in  which  the  first  members  are  perfect  squares.     This  is 

124.  It  in  reducing  an  equation  to  either  of  these  forms  the  co-effi- 
cient of  x1  is  negative,  what  do  you  do  ? 

125.  What  is  the  square  of  a  binomial  equal  to?  "What  does  the 
first  member  in  each  form  contain  ?  How  do  you  render  the  first  mem 
ber  a  perfect  square  ?     What  is  this  called  ? 


200  ELEMENTARY     ALGEBRA. 

called   completing    the   square.     Then,    by    extracting   the 
square  root  of  both  members  of  the  equation,  we  have 


x  +  p  —  ±  V?  +i>2> 
and  x  +  p  =  ±  -yj  — -  q  +  p2, 

which  gives,  by  transposing  />, 


a;  =  —  jp  ±  -y/—  g-  -f-^>2. 


126.  If  we  compare  the  second  and  fourth  forms  with 
the  square 

(a;  — p)2  =  x2  —2px  -f-  p2, 

we  also  see  that  half  the  co-efficient  of  x  being  squared  and 
the  result  added  to  both  members,  will  make  the  first  mem- 
bers perfect  squares.     Having  made  the  additions,  we  have 

x2  —  2px  +  p2  =  q  +  p2, 
x2  —  2px  -j-  p2  —  —  q  +  P2' 

Then,  by  extracting  the  square  root  of  both  members,  we 
have 


x—  p  =  ±^/q  +p2, 


and  x  —  p  =:  ±  -y/—  q  -+-  j>2 ; 

and  by  transposing     — p,   we  find 

x  =p  ±  y/q  +  ^2, 


and  a;  =^>  ±  y^  —  <?  -f-  p2- 


126i  In  the  second  form,  how  do  you  make  the  first  member  a  perfect 
square! 


EQUATIONS  OF  THE  SECOND  DEGREE.   201 

127.  Hence,  for  the  resolution  of  every  equation  of  the 
second  degree,  we  have  the  following 

RULE. 

I.  Reduce  the  equation  to  one  of  the  four  forms. 

II.  Take  half  the  co-efficient  of  the  second  term,  square  it% 
and  add  the  result  to  both  members  of  the  equation. 

III.  Then  extract  the  square  root  of  both  members  of  the 
equation  ;  after  which,  transpose  the  known  term  to  the  second 
member. 

Remark. — The  square  root  of  the  first  member  is  always 
equal  to  the  square  root  of  the  first  term,  plus  or  minus 
half  the  co-efficient  of  the  first  power  of  the  unknown 
quantity. 

EXAMPLES    OF    THE    FIRST    FORM. 

1.  What  are  the  values  of  x  in  the  equation 

2a;2  +  8x  =  64  ? 
If  we  first  divide  by  the  co-efficient  2  we  obtain 
x*  -V  4x  =  32. 
Then,  completing  the  square, 

x2  +  4x  +  4  =  32  4-  4  =  36. 
Extracting  the  root,  ** 

x  +  2  =  ±  ^30  =  +  6    or     —  6. 

Hence,  x  =  —  2  +  6  =  +  4  ; 

or,  x  =  —  2  —  6  =  —  8. 


127.  Give  the  general  rule  for  resolving  an  equation  of  the  second 
degree.  What  is  the  first  step  ?  What  the  second  ?  What  the  third ! 
What  is  the  square  root  of  the  first,  member  always  equal  to  ? 


202  ELEMENTARY      ALGEBRA. 

hence,  in  this  form,  the   smaller  root  is  positive,  and  the 
larger  negative. 

Verification. 

If  we  take  the  positive  value,  viz :  x  =  +  4, 

the  equation  x2  -f-  4*  =  32 

gives  42  -f  4  x  4  =  32  : 

and  if  we  take  the  negative  value  of  x,  viz  :  x  =  —  8, 

the  equation  x2  -f-  4x  =  32 

gives  (  -  8)2  +  4  (-  8)  =  04  -  32  =  32  ; 

from  which   we   see,   that   either  of  the  values  of  x,  vu 
x  =  +  4  or  x  =  —  8,  will  satisfy  the  equation. 

2.  What  are  the  values  of  x  in  the  equation 

3i-2  +  12a;  -  19  =  -  x2  -  12*  +  89  ? 
By  transposing  the  terms  we  have 

S*2  +  x2  +  12*  +  12*  =z  89  +  19  : 
and  by  reducing, 

4*2  +  24*  =  108  ; 
and  dividing  by  the  co-efficient  of  x2, 
x2  +  Gx  =  27. 
Now,  by  completing  the  square, 

x2  +  Gx  +  9  =  3G, 
oxtracting  the  square  root, 

x  +  3  =  ±  -/30  =  +  G  or  —  6  : 
hence,  x  =  +G  —  3=4-3; 

or.  x  =  —  6  —  3  =  —  9. 


EQUATIONS  OF  THE  SECOND  DEGREE.   203 

Verification. 

If  we  take  the  plus  root,  the  equation 
a2  +  Gx  =  27 
gives  (3)2  +  0  (3)  =  27 ; 

and  for  the  negative  root, 

x2  +  Gx  =  27 
gives         (-9)2  +  G(-9)  =  81 -54  =  27. 
4.  What  are  the  values  of  a;  in  the  equation 

x2  —  10*  +  15  =  %■  —  34a;  +  155. 
5 

By  clearing  of  fractions,  we  have 

5a;2  -  50a-  +  75  =  a'2  -  170a;  +  775 : 

by  transposing  and  reducing,  we  obtain 

4a'2  +  120a-  =  700  ; 

then,  dividing  by  the  co-efficient  of  a-2,  we  have 

x2  +  30a;  =  175  ; 

and  by  completing  the  square, 

a'2  +  30a;  +  225  =  400 ; 

and  by  extracting  the  square  root, 

x  +  15  =  ±  */400  =  +  20  or  —  20. 

Hence,  x  =  +  5  or  —  35. 

Verification. 
For  the  plus  value  of  x,  the  equation 
x2  +  30a;  =  175 
gives  (5)2  +  30  x  5  =  25  +  150  =  175. 


204  ELEMENTARY     ALGEBRA. 

And  for  the  negative  value  of  x,  we  have 

(  _  35)2  +  30(_  35)  —  1225  -  1050  =  175. 
5.  What  are  the  values  of  x  in  the  equation 

5       2  1  .      3  S  2  2  _l_    273  « 

Clearing  of  fractions,  we  have 

10a;2  -  Qx  +  9  =  96  —  8x  —  12x2  +  273  ; 

transposing  and  reducing, 

22z2  +  2x  =  360  ; 

dividing  both  members  by  22, 

.   .    2  360 

x*  -\ x  = . 

^22  22 

Add  ( —  l     to  both  members,  and  the  equation  becomes 

X  +  22  X  +  \22)         22  +V22j  ' 

whence,  by  extracting  the  square  root, 

1  /3G0    ,    /  1  \2 

a;  +  22=±V^r  +  l22J' 


therefore, 


1 
^-22  + 


360     ny 

22        \22/  ' 

,  1  /360    ,    /  1  \2 

and  •— S-Var.+  y- 


EQUATIONS  OF  THE  SECOND  DEGREE    205 

It  remains  to  perform  the  numerical  operations.     In  the 

860      /  1  \2 
first  place,  — - — h  I  — I    must  be  reduced  to  a  single  num- 
ber, having  (22)2  for  its  denominator. 

300       /J_\2_  360  x  22  +  1  _  7921 
'        22  +  V22/  ~  (22)2        ~  (22)2  i 

extracting  the  square  root  of  7921,  we  find  it  to  be  89 ; 
therefore, 


V    22   T  \22/  22 


Consequently,  the  plus  value  of  x  is 


J_   .   89       88 
22 

and  the  negative  value  is 


x  — 1 — —  4 

22  X  22  ~  22        ' 


X~  ~  22  +  22~~       11' 

that  is,  one  of  the  two  values  of  x  which  will  satisfy  the 
proposed  equation  is  a  positive  whole  number,  and  the  other 
a  negative  fraction. 

6.  What  are  the  values  of  x  in  the  equation 

3x2  +  2x  -  9  =  7Gfc 

A  i*=5 

Ans.  i  . 

Is  —  —  5f . 

7.  What  are  the  values  of  x  in  the  equation 

2a;2  4.  8*  +  7  =  ^  -  %■  +  197. 
4        8 


Ans-  IL-11 


3 


206  ELEMENTARY      &.LGEBRA. 


8.  What  are  the  values  of  x  in  the  equation 

£- -£-  +  15=-£-8*  +  95j. 
4         3  9 

Ans. 


9.  What  are  the  values  of  x  in  the  equation 
x2      5x  'x 

T~T~8=  2 


{x  =  0 
\x=  -  G4£. 

7x  +  G£. 


(  x  =  2 
Ans.   \ 

\  x  =  —  /£. 

10.  What  are  the  values  of  x  in  the  equation 

x2       x        x2       x       1 3 

2~  +  T  =  lf—  10  +  20' 

-**  i^=-2i. 

EXAMPLES    OF    THE    SECOND    FORM. 

1.  What  are  the  values  of  x  in  the  equation 
x2  —  8x+l0  =  19. 
By  transposing, 

x2  —  Sx  =  19-10  =  9, 
then  by  completing^Liie  square 

x2  —  8x  +  10  =  9  +  1G  =  25, 
and  by  extracting  the  root 

x  —  4  =  ±  -^/25  =  +  5     or     —  5. 
Hence, 

z  =  4  -f  5  =  9     or     z  =  4  —  5  =  —  1. 

Thal  is,  in  this  form,  the  larger  root  is  positive  and  the 
lesser  negative. 


EQUATIONS  OF  THE  SECOND  DEGREE.   207 

Verification. 

If  we  take  the  positive  value  of  x,  the  equation 

x2  —  8x  =  9     gives,     (9)2  —  8x9  =  81—  72  =  9; 

and  if  we  take  the  negative  value,  the  equation 

x2  -  Sx  =  9,     gives,     (  —  l)2  —  8(  —  1)  =  1+8  =  9; 

from  which  we  see  that  both  values  alike  saunsfy  the  equa« 
tion. 

2.  What  are  the  values  of  x  in  the  equation 

^  +  4-15  =  i-  +  2-  14}. 
2        o  4  * 

By  clearing  of  fractions,  we  have 

Gx2  +  4x—  180  =  Sx2  +  12a;  —  177, 
and  by  transposing  and  reducing 

Sx2  —  8a;  =  3, 
and  dividing  by  the  co-efficient  of  x2,  we  obtain 

2            8  1 

X1 —  X  =  1. 

o 

Then,  by  completing  the  square,  we  have 

2       8  16  16      25 

and  by  extracting  the  square  root, 


4         ,       A25 

5                5 
:  +  -^    or    --. 

Ilenoe, 

4        5 
*="2+-^=  +  3,    or 

4         5 
*-  3         3  - 

1 

3 

208  ELEMENTARY     ALGEBRA. 

Verification. 
For  the  positive  value  of  x,  the  equation 


x2 x  ■=.  1 

3 


gives  32--x3  =  9-8  =  l: 

and  for  the  negative  value,  the  equation 

2           8  1 

•      Xz —  X  =  1 

o 

/       "1  \2       8  1         1,8, 

glVes  (__j    __x-3=-  +  ¥  =  l. 

3.  What  are  the  values  of  x  in  the  equation 

f —  4-73  —at 

2        3  +    8 

Clearing  of  fractions,  and  dividing  by  the  co-efficient  of 
a?2,  we  have 

x2-—x  =  \\. 

o 

Completing  the  square,  we  have 

X       JX+  9  -1**  9-36' 
then,  by  extracting  the  square  root,  we  have 

1  /49        ,    7  7 

x  ~  ~  -  =fc  V  W-  -  +  T'    or    —  T » 
o  V    ob  b  0 


hence, 


1,7        9        n  17  5 


EQUATIONS     OF     THE     SECOND     DEGREE..      209 

Verification. 
If  we  take  the  positive  value  of  x,  the  equation 

x2  —  —  x  =  \\ 

gives,  (U)2~|  XlA  =  2i-l  =  li: 

and  for  the  negative  value,  the  equation 

x2  —  —  x  —  1  i 

/       5  \2       2  5       25   ,    10       45       _ , 

glves,  (-_)  __x__  =  _  +  _  =  _  =  ii. 

4.  What  are  the  values  of  x  in  the  equation 
4a2  —  2x2  +  2ax  =  18ab  —  1S62  1 

By  transposing,  changing  the  signs,  and  dividing  by  2,  the 
equation  becomes 

x2  —  ax  =  2a2  —  9ab  +  9b2  ; 
whence,  completing  the  square, 

ft  Mf1/ 

x2  _  ax  +  "   _  » 9a5  +  952  . 

4  4 

extracting  the  square  root, 

x  =  ^±x/^~-9ab  +  9b2. 
2       V    4 

9a2 
Now,  t^e  square  root  of   — 9ab  -f-  9J2,    is    evidently 

^  —  36.     Therefore, 

a        /3a       „,  \  {z=       2a  —  36 

•T*(t-*»)  or      1*=-   a  +  U. 

10 


210 


ELEMENTARY     ALGEBRA. 


What  -will  be  the  numerical  values  of  x:  if  we  suppose 
0  =  6  and  b  =  1  % 

5.  What  are  the  values  of  x  in  the  equation 

1  4 

—  x  —  4  —  x2  +  2x  —  —  x1  =  45  —  3x2  +  Ax  ? 
3  5 


v4ns. 


»  =       7.! 2  )  to  within 
a?  =—5.73)       0.01. 
C.  What  are  the  values  of  x  in  the  equation 
8x2  —  14x  +  10  =  2x  4-  34  % 

Ans. 


{x=       3. 
\x  =  —  1. 


7.  What  are  the  values  of  x  in  the  equation 


—  30  4-  x  =  2x  —  22  ? 


8.  What  are  the  values  of  a;  in  the  equation 

x1 
x2-3x  +  —  =  9z+  13?  ] 


-4  ns. 

9.  What  are  the  values  of  a:  in  the  equation 

2ax  ~x2  —  —2ab  —  b2l 

Ans.   \ 

(  x  = 

10.  What  are  the  values  of  a;  in  the  equation 


(*==       8. 
*  \  x  =  —  4. 


*  =       9. 
s  =  -  1. 


2a  +  J. 


a2  +  £2  —  2bx  +  x2  = 


Ans. 


I  bn  -f  y  a5 


//i2  +  &2W42 


uhA. 


—  n2  —  mT  \hn  ~  Va2/«2  +  b2™2  —  a2n2\ 


EQUATIONS     OF     THE     SECOND     DEGREE.      211 
EXAMPLES    OF    THE    THIRD    FORM. 

1.  What  are  the  values  of  x  in  the  equation 

x2  +  4ar  =  —  3  1 
First,  by  completing  the  square,  we  have 
a;2  +  4ar  +  4  =  —  3  +  4=1  ; 
and  by  extracting  the  square  root, 

x  +  2  =  ±  yT=  +  1,    or,    -  1  : 
hence,    ar  =  —  2  +  1  =  —  1;    or   x  =.  —  2  —  1  =  —  3* 
That  is,  in  this  form  both  the  roots  are  negative. 

Verification. 

If  we  take  the  first  negative  value,  the  equation 
x2  +  4x=  —  3 
gives,  (-l)2  +  4(-rl)  =  l-4=-3; 

and  by  taking  the  second  value,  the  equation 

x2  +  4x  =  —  3. 
gives,  (-3)2  +  4(-3)  =  9  — 12  =  —3: 

hence,  both  values  of  a:  satisfy  the  given  equation. 

2.  What  are  the  values  of  x  in  the  equation 

—  ~  -  5x  —  1G  =  12  +  4-z2  +  Gar. 

By  transposing  and  reducing,  we  have 
—  x2  —  liar  =  28; 

then  since  the  co-efficient  of  the  second  power  of  x  is  nega- 
tive,  we  change  the  signs  of  all  the  terms,  which  gives 

a:2  +  llar  =  -28, 


412  ELEMENTARY    ALGEBRA. 

then  by  completing  the  square 

x2  +  11x4-30.25  =  2.25, 
hence, 

x  +  5.5  =  ±  -v/2.25  =  -f-  1.5     or     —1.5; 

consequently, 

x  =  —  4     or     x  =  —  7. 
3.  What  are  the  values  of  x  in  the  equation 


—  —  —  2x  —  5  =  —  x2  +  5x  +  5. 


(  x  = 


-5 


(  a;  = 


4.  What  are  the  values  of  x  in  the  equation 

2x2  +  8x  =  -2§  — -^-x. 

o 

-4. 

-£• 

5.  What  are  the  values  of  x  in  the  equation 

q 

4x2  +  —  x  +  ox  =  —  14a  —  S\  —  4x2. 
5 

(  x  =  o# 

(  z  -  -  ± 

6.  What  are  the  values  of  x  in  the  equation 

3  4x2 

—  x2  —  4  —  —  x  =  —  +  24x  +  2. 


8. 
7.  What  are  the  values  of  x  in  the  equation 


Ans.   \ 

(  x  =  — 


4-x2  +  ?x  +  20  =  -  ^-x2  -  llx  -  60. 


Ans.  ] 

(  x  =  —    8. 


EQUATIONS  OF  THE  SECOND  DEGREE.   213 

8.  What  are  the  values  of  x  in  the  equation 
5    .  .1  „,         1     „       1 


_  x* ._  x  +  _  _  _  Qix  _  _  X2  _  _. 


Ans. 


j  ar  =  — 8 
(  x=  -h 


9.  What  are  the  values  of  x  in  the  equation 


4      2    .     r        ,      1  1       2  r   1  3 

-  z2  +  5x  +  -  =  -  -  z2  -  5TL*  -  - 


j  x=  —  10 

10.  What  are  the  values  of  x  in  the  equation 

g  —  z2  —  3  =  Gx  -f  1. 

4u   i*=-4 
(  x=  -1 

11.  What  are  the  values  of  x  in  the  equation 

z2  +  4*  -  90  =  —  93. 

.         i  x—  —  3 
Am'  \  x  =  -  1. 

EXAMPLES  OF  THE  FOURTH  FORM. 

1.  What  are  the  values  of  x  in  the  equation 
x°~  —  Sx=  —  7. 
By  completing  the  square  we  have 

.r2  -  8x  +  16  =  —  7  +  10  =  9 ; 
then  by  extracting  the  square  root 

x  —  4=±yr9=+3    or     -3; 
hence, 

x  =  -f-  7    or    x  =  4-  1. 

That  is,  in  tills  form,  both  the  roots  are  positive. 


214  ELEMENTARY     ALGEBRA. 

Verification. 

If  we  take  the  greater  root,  the  equation 
x2  -  8x  =  —  7     gives     ?2  -8x7=  49-50  =  -  7  • 
and  for  the  less,  the  equation 

a.-2-8x=— 7     gives     P-8  X  1  =  1  -8=  -?. 
hence,  both  of  the  roots  will  satisfy  the  equation. 

2.  What  are  the  values  of  x  in  the  equation 

40 
-  ll*2  +  Sx  -  10  =  \\x2  -  IS*  +  -^ . 

Cy  clearing  of  fractions  we  have 

—  3x2  +  Gx  —  20  =  Sx2  —  2Qx  4-  40  ; 

then  by  collecting  the  similar  terms 

—  Gx2  4-  42*  =  GO  ; 

then  by  dividing  by  the  co-efficient  of  x2,  and  at  the  same 
time  changing  the  signs  of  all  the  terms,  we  have 

x2  —  7x  =  —  10. 

By  completing  the  square,  we  have 

X2  _  7s  _}_  X2.25  =  2.25, 

and  by  extracting  the  square  root  of  both  members, 

x-3.5  =  ±vA^25  =  +  1.5    or     -1.5; 
hence, 

x  =  3.5  +  1.5  =  5,     or     x  —  3.5  —  1.5  =  2. 


EQUATIONS     OF     THE     SECOND     DEGREE.      215 

Verification. 

If  we  take  the  greater  root,  the  equation 
z2-7x=-10     gives     52  -  7  x  5  =  25  —  35  =  —  JO  ; 
and  if  we  take  the  lesser  root,  the  equation 
a;2  — 7a;  =  —  10     gives     22  —  7  X  2  =  4  -  14  =  -  10. 

3.  What  are  the  values  of  x  in  the  equation 
—  3a;  +  2a:2  4-  1  =  17|-x  -  2a;2  —  3. 
By  transposing  and  collecting  the  terms,  we  have 

4a;2  —  20  jx  =  —  4  ; 
then  dividing  by  the  co-efficient  of  x2,  we  have 

x2  —  5\x  =  —  1. 
By  completing  the  square,  we  obtain 

,       r.         109  ,    ,   109       144 

*   -5^+25-=-1+25T=25-' 
and  by  extracting  the  root 


/144        ,    12 
hence, 


12 

— --; 


12  12        1 

5   _  o  ,    or,     x  _  Z&  _   5   _  5 


Verification. 

If  we  take  the  greater  root,  the  equation 
x1  —  5|ar  =  —  1,    gives,    52  —  5]  X  5  =25— 26  =  —  1. 

and  if  we  take  the  lesser  root,  the  equation 

,        n  ,        •  /  1    \2        n  1  1  26 

*-5l*  =  -l,  glves,(yj-5lx-  =  25-^=-l 


216  ELEMENTARY      ALGEBRA. 

4.  What  are  the  values  of  x  in  the  equation 

1        2  o      _,        1  6       2     ,       1  l    7 

Ans. 

5.  What  are  the  values  of  x  in  the  equation 

—  4x2  —  —  x  +  11  =  —  5*2  +  8*  ? 


«  =  3. 


x  —  Y, 


6.  What  are  the  values  of  x  in  the  equation 

'  2 


X   -E 


7.  What  are  the  values  of  x  in  the  equation 

x2-l0^x=  —  1? 

At 

8.  What  are  the  values  of  a;  in  the  equation 


Ins.     < 


1 7a;2  2.x2 

21x  +  — -  +  100  =  —  +  12*  -  2G  ? 
5  5 


(a:  =  G. 


9.  What  are  the  values  of  x  in  the  equation 

8x2  llx2 

22x  +  15  =  -  -—  +  28x  -  30? 

o  o 

-4  ns. 
10.  What  are  the  values  of  x  in  the  equation 

o 

2x2  -  30x  -1-  3  =  -  x2  +  3^a;  -  —  ? 


x  =  9 

x  =  l 


,  x  =  11 
./l«s.     •(  j 

iff- 


EQUATIONS  OF  THE  SECOND  DEGREE.   217 

Properties  of  the  Boots. 

128.  We  have  thus  far,  only  explained  the  methods  of 
inding  the  roots  of  an  equation  of  the  second  degree.  We 
are  now  going  to  show  some  of  the  properties  of  these  roots. 

First  form. 

129.  In  the  first  form 

x2  +  2}?x  =  q ; 
hence,   1st  root  z  =  —  p  -f  -y/q  -f  p2\ 

2d  root  x  =  —  p  —  i/q  -f  p2, 

and  their  sum  =  —  2p. 

Since,  in  this  form  q  is  supposed  positive,  the  quantity 
q  +  p>2  under  the  radical  sign  will  be  greater  than  p2,  and 
hence  its  root  will  be  greater  than  p.  Consequently,  the 
first  root,  which  is  equal  to  the  difference  between  p  and 
the  radical,  will  be  positive  and  less  than  -y/q  -f-  p2.  In  the 
second  root,  p  and  the  radical  have  the  same  sign ;  hence, 
the  second  root  will  be  equal  to  their  sum,  and  negative. 
If  we  multiply  the  two  roots  together,  we  have 

— p  +  Vq  +  P2 

—  P    —    Vfl  +  P 


+  p2  —  py/q  +  p2 

+  Py'q  +  p  —  q—p2 

Product  equal  to —  q. 


129.  In  the  first  form,  have  the  roots  the  same  or  contrary  signs? 
What  is  the  sign  of  the  first  root  ?  What  of  the  second  ?  Which  is 
the  greater!  What  is  their  sum  equal  tot  What  is  their  prod'v 
equal  to? 

10 


218  ELEMENTARY      ALGEBRA. 

Hence  we  conclude, 

1st.  That  in  the  first  form,  one  of  the  roots  is  always  posi 
live  and  the  other  negativt 

2d.  That  the  positive  root  is  numerically  less  than  the 
negative  root. 

3d.  That  the  sum  of  the  two  roots  is  equal  to  the  co-efficient 
of  x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.  Tliat  the  product  of  the  two  roots  is  equal  to  the 
second  member,  taken  with  a  contrary  sign. 

EXAMPLES. 

1.  In  the  equation 

x2  +  x  =  20, 

we  find  the  roots  to  be  4  and  —  5.     Their  sum  is  —  1,  and 

their  product  —  20. 

2.  In  the  equation 

x2  +  2x  =  3, 

we  find  the  roots  to  be  1  and  —  3.    Their  sum  is  equal  to 
—  2,  and  their  product  to   —  3. 

3.  The  roots  of  the  equation 

x2  +  x  =  90, 

are  -J-  9  and  —  10.     Their  sum  is  —  1,  and  their  product 
-90. 

4.  The  roots  of  the  equation 

x2  +  4^=00, 

are  6  and  —  10.     Their  sum  is  —  4,  and  their  product  Is 
-GO. 


EQUATIONS     OF     THE     SECOND     DEGREE.       219 

Let  these  principles  be  applied  to  each  of  the  examplea 
under  "  examples  of  the  first  form." 


Second  Form. 

130.  The  second  form  is, 

x2  —  2px  =  q  ; 

and  by  resolving  the  equation  we  find 

1st  root,  x  =  -f  p  +  V  Q  4  P2 

2d  root,  x  =  +  p  —  sj  q  +  p2, 

and  their  sum  =  2p. 

In  this  form,  the  first  root  is  positive  and  the  second 
negative.     If  we  multiply  the  two  roots  together,  we  have 

(*'+  Vs  +  p2)  x(p-  V9+p2)  =  —  q- 

Hence,  we  conclude, 

1st.  That  in  the  second  form,  one  of  the  roots  is  positive 
and  the  other  negative. 

2d.  That  the  positive  root  is  numerically  greater  than  the 
negative  root. 

3d.  That  the  sum  of  the  roots  is  equal  to  the  co-efficient  of 
x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.  That  the  product  of  the  roots  is  equal  to  the  second 
member,  taken  with  a  contrary  sign. 


130.  What  is  the  sign  of  the  first  root  in  the  second  form  ?  "What  is 
the  sign  of  the  second  ?  Which  is  the  greater  ?  What  is  their  sum 
equal  to  ?     What  is  their  product  equal  to  f 


2&0  ELEMENTARY      A  h  G  E  V  V  A  . 

EXAMPLES 

1.  The  roots  of  the  equation 

x2  —  x  =  12, 
are  -j-  4  and  —  3.     Their  sum  is  +  1,  and  their  product 
-12. 

2.  The  roots  of  the  equation 

*  -  9TVs  =  1, 

1 

are  +10  and  —77:-     Their  sum  is  9  A,  and  their  product 
10 

is  —  1. 

3.  The  roots  of  the  equation 

x2  —  Gx  —  10, 
are   +  8  and  —  2.     Their  sum  is  +  0,  and  their  product 
is  —  1G. 

4.  The  roots  of  the  equation 

x2  —  Ux  =  80, 
are  +10  and  —  5.     Their  sum  is  +  11,  and  their  product 
is  —80. 

Let  these  principles  he  applied  to  each  of  the  examples 
under  "  examples  of  the  second  form." 

Third  Form. 

131.  The  third  form  is, 

x2  +  2px  =  —  q  ; 
and  by  resolving  the  equation,  we  find, 

1  st  root,  x  =  —  p  +  -y/  —  q  +  jr;2, 


2d  root,  x  =  —  p  —  -yj  —  q  +  jt>2 


and  their  sum  is       =  —  2p. 


EQUATIONS  OF  THE  SECOND  DEGREE.   22] 

In  this  form,  the  quantity  under  the  radical  being  less 
than  2>2,  its  root  will  be  less  than  p  :  hence,  both  the  roots 
will  be  negative,  and  the  first  will  be  numerically  the  least. 

If  we  multiply  the  roots  together,  we  have 

(~P  +  V  -  9  +  P2)  X  (  - p  -  V  -  q  +  p2)  =  +  q. 

Hence,  we  conclude, 

1st.   That  in  the  third  form  both  the  roots  are  negative. 

2d.   That  the  first  root  is  numerically  less  than  the  second. 

3d.  That  the  sum  of  the  two  roots  is  equal  to  the  co-efficient 
of  x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.  That  the  product  of  the  roots  is  equal  to  the  second 
member,  taken  with  a  contrary  sign. 

EXAMPLES. 

1.  The  roots  of  the  equation 

x2  +  dx  =  —  20, 

are  —  4  and  —  .5.     Their  sum  is  —  9,  and   their  product 
+  20. 

2.  The  roots  of  the  equation 

x2  +  Ux  =  —42, 

are  —  G  and  —  7.     Their  sum  is  —  13,  and  their  product 
+  42. 


131.  In  the  third  form,  what  are  the  signs  of  the  roots  ?  Which  root 
is  the  least  ?  What  is  the  sum  of  the  roots  equal  to  ?  What  is  their 
product  equal  to  ? 


222  ELEMENTARY     ALGEBRA. 

3.  The  roots  of  the  equation 

3 

are  —  —  and  —  2.     Their  sum  is  —  2|,  and  their  product 

+  IJ. 

4.  The  roots  of  the  equation 

x2  +  5x  =  —  C, 

are  —  2  and  —  3.     Their  sum  is  —  5,  and  their  product 
is  +  G. 

Let  these  principles  be  applied  to  each  of  the  examples 
under  "  examples  of  the  third  form." 

Fourth  Form. 

132.  The  fourth  form  is, 

x2  —  2px  ==  —  q  ; 
and  by  resolving  the  equation  we  find, 


1st  root,  x  =  p  +  -\/  —  q  +2^ 


2d  root,  x  =  p  —  y/  —  ^  +P5 

Their  sum  is  =  2p. 

In  this  form,  as  well  as  in  the  third,  the  quantity  under 
the  ladical  sign  being  less  than  p2,  its  root  will  be  less  than 
p :  hence  both  the  roots  will  be  positive,  and  the  first  will 
be  the  greater. 

If  we  multiply  the  two  roots  together,  we  have 
(p  +  V~g+P2)  X(p-  V  -q+p2)  =  +y. 


EQUATIONS  OF  THE  SECOND  DEGREE.   223 

Hence  we  conclude, 

1st.    That  in  the  fourth  form,  both  the  roots  are  positive. 

2d.    That  the  first  root  is  greater  than  the  second. 

3d.  That  the  sum  of  the  roots  is  equal  to  the  co-efficient  of 
x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.  That  the  product  of  the  roots  is  equal  to  the  second 
member,  taken  with  a  contrary  sign. 

EXAMPLES. 

1.  The  roots  of  the  equation 

x2  -  Ix  =  -  12, 

are  +  4  and  +  3.     Their  sum  is  -f-  7  and  their  product 
+  12. 

2.  The  roots  of  the  equation 

x2  -  14x  =  -  24, 

are  +12  and  +  2.     Their  sum  is  +  14  and  their  product 
1-24. 

3.  The  roots  of  the  equation 

x2  —  20x  =  —  36, 

are  +18  and  +  2.     Their  sum  is  +  20  and  their  product 
+  3G. 

4.  The  roots  of  the  equation 

x1  —  \lx=—  42, 
are  +  14  and  +  3.     Their  sum  is  +  17  and  their  product 
f  42. 

132.  In  the  fourth  form,  -what  are  the  signs  of  the  roots  ?  Which  root 
id  the  greater  ?  What  is  the  sum  of  the  roots  equal  to  ?  What  is  their 
Vroduct  equal  to  S 


224  ELEMENTARY    ALGEBRA. 

133.  In  the  third  and  fourth  forms  the  values  of  x  some- 
times become  imaginary,  and  in  such  cases  it  is  necessary 
to  know  how  the  results  are  to  be  interpreted. 

If  we  have  q  >  p2,  that  is,  if  the  second  member  is  greaier 
than  half  the  co-efficient  of  x  squared,  it  is  plain  that  y — q+p2 
will  be  imaginary,  since  the  quantity  under  the  radical  sign 
will  be  negative.  Under  this  supposition  the  values  of  x, 
in  the  third  and  fourth  forms,  will  be  imaginary. 

We  will  now  show  that,  when  in  the  third  and  fourth 
forms,  we  have  q  >  p2,  the  conditions  of  the  problem  will 
be  incompatible  with  each  other. 

134.  Before  showing  this  we  will  demonstrate  a  proposi- 
tion on  which  the  proof  of  the  incompatibility  depends  :  viz. 

If  a  given  mirnber  be  decomposed  into  two  'parts  and  those 
parts  multiplied  toe/ether,  the  product  will  be  the  greatest  pos- 
sible when  the  parts  are  equal. 

Let  2/>  be  the  number  to  be  decomposed,  and  d  the  dif- 
ference of  the  parts.     Then 

p  +  —  =  the  greater  part  (page  104,  Ex.  7.) 

lit 

and        p   —  —  =         the  less  part ; 

d2 
and       p1 —  =  P,     their  product  (Art.  40.) 

Now,  it  is  plain  that  P  will  increase  as  d  diminishes,  and 
that  it  will  be  the  greatest  possible  v,  hen  d  =  0  :  that  is, 
p  X  P  =  p2     is  the  greatest  product. 

1 33.  In  which  forms  do  the  values  of  x  become  imaginary?  When 
will  the  values  of  x  be  imaginary?  Why  will  the  .values  of  x  be  tlicD 
imaginary  ? 


EQUATIONS     OF     THE     SECOND     DEGREE.      225 

Now,  since  in  the  equation 

x2  —  2px  =  —  q 

l2p  is  the  sura  of  the  roots,  and  q  their  product,  it  follows 
that  q  can  never  be  greater  than  p2.  The  conditions  of  the 
proposition,  therefore,  fix  a  limit  to  the  value  of  q,  and  if 
we  make  q  >  p2,  we  express  by  the  equation  a  condition 
which  cannot  be  fulfilled,  and  this  impossibility  is  made 
apparent  by  the  values  of  x  becoming  imaginary.  Hence, 
we  may  conclude  that, 

When  the  values  of  the  uriknoivn  quantity  are  imaginary, 
the  conditions  of  the  proposition  are  incompatible  with  each 
other. 

EXAMPLES. 

1.  Find  two  numbers  whose  sum  shall  be  12  and  pro- 
duct 46. 

Let  x  and  y  be  the  numbers. 

By  the  1st  condition,     x  +  y  =  12 ; 

and  by  the  2d,  xy  =  46. 

The  first  equation  gives 

x  =  12  —  y. 

Substituting  this  value  for  x  in  the  second,  we  have 

\2y  -  y2  =  4G  ; 

and  changing  the  signs  of  the  terms,  we  have 

y2  —  12y  =  —46. 


134.  What  is  the  proposition  demonstrated  in  Article  134?  If  the 
conditions  of  the  question  are  Incompatible,  how  will  the  values  of  the 
unknown  quantity  be  J 

10* 


220  ELEMENTARY     ALGEBRA. 

Then,  by  completing  the  square 

y9-  —  12y  +  30  =  —  40  +  30  =  —  10 

which  gives  y  =  0  +  y'  —  10, 

and  y  —  0  —  ^  —  10  ; 

both  of  which  values  are  imaginary,  as  indeed  they  should 
be,  since  the  conditions  are  incompatible. 

2.  The  sum  of  two  numbers  is  8,  and  their  product  20  ' 
what  are  the  numbers  ? 

Denote  the  numbers  by  x  and  y. 
By  the  first  condition, 

z  +  y  =  S; 
and  by  the  second,  xy  =  20. 

The  first  equation  gives 

x  =  8  —  y. 
Substituting  this  value  of  x  in  the  second,  we  have 
8y  -  2/2  =  20  ; 
changing  the  signs,  and  completing  the  square,  we  have 

y*  _  8y  +  10  =  -  4 ; 
and  by  extracting  the  root, 

y  =  4  +  V  —  4   and   y  =  4  —  -y/  —  4. 

These  values  of  y  may  be  put  under  the  forms  (Art.  106) 

y  =  4  +  2  V  —  1    aud    y  =  4  —  2  -v/^TL 

3.  What  are  the  values  of  x  in  the  equation 

x2  +  2x=  —  10. 

<r  =  —  1  +  3  i/—"- 


,4ns. 

'  *=  -1  -3-i/^TI. 


EQUATIONS     Of'tHE     SECOND     DEGREE.       227 

Examples  invoicing  more  titan  one  unlcnown  quantity. 

1.  Given       \  "  y     to  find  x  and  y. 

I  x2+  y2  =  100  \ 

By  transpusing  y  in  the  first  equation,  we  have 

X  =  14  —  y  ; 

and  by  squaring  both  members, 

a-2  =  100  —  2Sy  +  y\ 

Substituting  this  value  for  x2  in  the  2d  equation,  we  have 
190  —  28y  +  y2  +  y2  =  100  ; 
from  which  we  have 

y2_14y=_48. 

and  by  completing  the  square, 

V2  -  14y  +  49  =  1  ; 
and  by  extracting  the  square  root, 

y  -  7  =  =b  -/T  =4-1    or    -  1 ; 

hence,         y  =  7  +  1  =  8,    or    y  =  7  —  1  =  G. 

If  we  take  the  greater  value,  we  find  x  =  G  ;  and  if  we 

take  the  lesser,  we  find  x  =  8. 

Verification. 

For  the  greater  value,  y  =  8,  the  equation 

x  +  y   =    14     gives       6  +    8  =:  14  ; 
and  z2  +  y2:=100     gives     3G  +  G4  =  100. 

For  the  value  y  =.  (S,  the  equation 

x  +  y   =    14     gives       8  +    6  =    14 ; 
and  x2  4-  y2  =  100     gives     G4  4-  30  =  100. 

Hence,  both  sets  of  values  will  satisfy  the  given  equation. 


228  ELEMENTARY     ALGEBRA. 

2.  Given     •]       —  "   ~        >    to  find  x  and  y. 

(  x2  —  y2  =  45  ) 

Transposing  y  in  the  first  equation,  we  have 

and  then,  squaring  both  members, 

x2  =  9  +  6y  +  y2. 

Substituting  this  value  for  x2,  in  the  second  equation,  we 
have 

9  +  0y  +  y2-y2  =  45; 
whence  we  have 

Cy  =  30     and     y  —  6. 
Substituting  this  value  of  ?/,  in  the  first  equation,  we  have 
x  —  6  =  3, 
and  consequently         a:  =  3  +  0  =  9. 

a;  —  y  =  3     gives     9  —  6  =  3; 
and  x2  —  y2  =  45     gives     81  —  30  =  45. 

n     _.  (  z2  +  3.ry  =  22  )     .     G    ,  , 

3.  Given     \        \       y    ,  5-    to  find  a;  and  y. 

(  a;2  +  3a;y  +  2y2  =  40  ) 

Subtracting  the  first  equation  from  the  second,  we  have 
2y2  =  18, 
which  gives  y2  =  9, 

and  y  =z  +  3,  or  —  3. 

Substituting  the  plus  value  in  the  first  equation,  we  have 
x2  +  9x  =  22  ; 


EQUATIONS     OF     THE     SECOND     DEGREE.      229 

from  which  we  find 

x  =  +  2     and     x  =  —  11. 

If  we  take  the  negative  value,  y  =  —  3,  we  have  from  the 
first  equation, 

x2  —  9x  =  22  ; 
from  which,  we  find 

x  =  +  1 1     and    x  =.  —  2. 

Verification. 

For  the  values  y  =  +  3  and  x  =  +  2,  the  given  equation 

z2  +  3zy  =  22 
gives  22 +  3x2x3  =  4+ 18  =  22; 

and  for  the  second  value,  x  =  —  11,  the  same  equation 

x2  +  3ary  =  22 

gives,     (  -  11)2  +  3  x  _  n  x  3  =  121  -  99  =  22. 

If  now  we  take  the  second  value  of  y,  that  is,  y  —  —  3, 
and  the  corresponding  values  of  x,  viz.,  x  =  +  11,  and 
a:  =  —  2  ;  for  #=  +  11,  the  given  equation 

x2  +  3xy  =  22 
gives,       ll2  +  3x  11  X —3  =  121 —99  =  22; 
and  for  x  =  —  2,  the  same  equation 

x2  +  3xy  =  22 
gives,       (  -  2)2  +  3  x  -  2  x  -  3  =  4  +  18  =  22. 

f  xz  =  y2     (1)  \ 

4.  Given    }  x  +  y   +  z  =    7     (2)  >    to  find  #,  y,  and  2. 
(  x2  +  y2  +  z2  =  21     (3)  ) 


230  ELEMENTARY     ALGEBRA. 

Transposing  y  in  the  second  equation,  we  have 
x+z=7-y     (4); 
then  squaring  the  members,  we  have 

x2  +  2xz  +  z2  =  49  —  14y  +  y2. 

If  now  we  substitute  for  2xz  its  value  taken  fiom  the 
first  equation,  we  have 

X2  _j_  0//2  4.  Z2  _  49  _  14y  +  y2  . 

and  cancelling  ?/2,  in  each  member,  there  results 

x1  4-  y2  +  £2  =  49  —  14y. 

But,  from  the  third  equation  we  see  that  each  member  Oi 
the  last  equation  is  equal  to  21  :  hence 

49-14y  =  21, 

and  14y  =  49  —  21  =  28  ; 

28 
hence,  y  =  —  =  2. 

Placing  this  value  for  y  in  equation  (1),  gives 

xz  =  4 ; 
and  placing  it  in  equation  (4),  gives 

x  +  z  =  5,     and     x  =  5  —  z. 

Substituting  this  value  of  x  in  the  previous  equation,  we 
obtain 

52  —  z2  =  4,     or     s2  —  5z  =  —  4  ; 

and  by  completing  the  square,  we  have 

22  _  5z  4.  G.25  =  2.5, 
and        2  —  2.5  =  =fc  -/23"=  +  1.5     or     —  1.5  ; 
hence,  z  =  2.5  -I-  1.5  =  4    or    c  =  -f-  2.5  -  1.5  =  1. 


EQUATIONS  OF  THE  SECOND  DEGREE.   231 

If  we  take  the  value 

2  =  4,     we  find     x  =  1  : 
if  we  take  the  lesser  value 

z  =  1,     we  find     x  =  4. 

3.  Given     x  +  y/~xy  -j-  y   =    Id   {         r  , 

„  o       -.oo   f    to  find  r  and  y. 

and         #2  +       a-y  +  2/   —  !**«*   ' 

Dividing  the  second  equation  by  the  first,  we  have 
x—  -/jy  4-    y  =    7 

but,  £  +  V^  +    V  =  1& 

hence,  by  addition,  2x  -f-  2y  =  2G 

or  «+    y  =  13 

and  substituting,  in  lstequa.  -\J '  xy-\-  13  =  19 
or,  by  transposing  -y/xy=    G 

and  by  squaring  xy  =  3G. 

Equation  2d,  is  x2  +  xy  4-  y2  —  133 

and  from  the  last,  we  have  3.ry  —  108. 

Subtracting  a:2  —  2xy  4-  y2=    25 

hence,  a;  —  y  =  ±      5 

but  x  4-  y  =         13 

hence  a?  =  9  or  4 ;    and    y  =  4  or  9. 

0.  Given  the  sum  of  two  numbers  equal  to  cr,  and  the 
sum  of  their  cubes  equal  to  c,  to  find  the  numbers. 


By  the  conditions  "1    ,   .     , 

•>  (  a-3  +  y3  =  c. 


232  ELEMENTARY     ALGEBRA. 

Putting       x  *=  s  -f  z,     and     y  =  s  —  z,     we  have 

a 

a  =  2s,     or     s  =  —  5 


I 


2 

Z3   =  S3  +  3*23  +  3S22  +  Z3 

v3  =  s3  —  3s2z  4-  3sz2  —  z3. 


hence,  by  addition,       i3  +  2/3  =  2s3  +  Qsz2  =  c, 

whence,  z2  =  — - —    and    z  =  ±  \  / 

'  6s  V 


C  —  2a 


6s 


/c-2s3  ,  /c  —  2s* 

or,       .^.d-yT--—.;    and  y^s^  — _; 

or,  by  putting  for  s  its  value, 
_o3 


4c  —  a3 


12a 


2       V   \    3a    /        2+V      12a 

Note. — What  are  the  numbers  when  a  =  5  and  c  '-=■  35. 
What  are  the  numbers  when  a  =  9  and  c  =  243  1 


QUESTIONS. 

1.  Find  a  number  such,  that  twice  its  square,  added  to 
three  times  the  number,  shall  give  65. 

Let  x  denote  the  unknown  number.     Then  the  equation 
of  the  problem  will  be 

2x2  +  Sx  =  G5, 
whence, 

Z~"4±V2"  +  lG~~4        4* 


EQUATIONS  OF  THE  SECOND  DEGREE.   233 

Therefore. 

*=__+T==5,     and     *  =  ____.  =  __-. 

Both  these  values  satisfy  the  proposition  in  its  algebraic 
sense.     For, 

2  X  (5)2  +  3x5=2x25  +  15=65; 

.nd    2(-_)   +3x--  =  — --  =  — =65. 

Remark. — If  we  wish  to  restrict  the  enunciation  to  its 
arithmetical  sense,  we  will  first  observe,  that  when  x  is  re- 
placed by  —  x,  in  the  equation  2.r2  +  ox  =  G5,  the  sign  of 
the  second  term  3.c  only,  is  changed,  because  (  —  x)2  =  x2. 

3  23 

Therefore,  instead  of  obtaining  x  = ±  — ,    we  should 

4  4 

3  23  13  *       i 

find  x  —  ~r  =b  -7-5    or  a;  =  -— ,    and  a;  =  —  5,  values  which 

4  4  2 

only  differ  from   the  preceding  by  their  signs.     Hence,  we 

13 
may  say  that  the  first  negative  result, — ,    considered  in- 
dependently of  its  sign,  satisfies  this  new  enunciation,  viz: 

To  find  a  number  such,  that  twice  its  square,  diminished 
by  three  times  the  number,  shall  give  65.     In  fact,  we  have 


(¥)' 


,  13  \a      „       IS       169      30       nr 
-)-3X-  =  — --  =  65. 


Remark. — The  root  which  results  from  giving  the  plus 
sign  to  the  radical,  is,  generally,  an  answer  to  the  question 
both  in  its  arithmetical  and  algebraic  sense  ;  while  the  second 
root  is  an  answer  to  it  in  its  algebraic  sense  only. 
11 


234  ELEMENTARY     ALGEBRA. 

Thus,  in  the  example,  it  was  required  to  find  a  number, 
of  which  twice  the  square  added  to  three  times  the  number 
shall  give  65.  Now,  in  the  arithmetical  sense,  added  means 
increased  ;  but  in  the  algebraic  sense  it  implies  diminution, 
when  the  quantity  added  is  negative.  In  this  sense,  the 
second  root  satisfies  the  enunciation. 

2.  A  certain  person  purchased  a  number  of  yards  of  cloth 
for  240  cents,  if  he  had  received  3  yards  less  of  the  same 
cloth  for  the  same  sum,  it  would  have  cost  him  4  cents  more 
per  yard.     How  many  yards  did  he  purchase  1 

Let     x  —     the  number  of  yards  purchased. 

240 
Then    will  express  the  price  per  yard. 

If,  for  240  cents,  he  had  received  3  yards  less,  that  is 

x  —  3  yards,  the  price  per  yard,  under  this  hypothesis,  would 

240 
have  been  represented  by     -.     But,  by  the  enunciation, 

X  —  O 

this  last  cost  would  exceed  the  first  by  4  cents.     Therefore, 
we  have  the  equation 

240        240 

x—2         x ~     ' 
whence,  by  reducing,    x%  —  Zx  =  180, 


3  /9         ,™      3  ±27 

and  a==_±v/_.  4-180  =  —^—; 

therefore,  x  =  15     and     x  —  —  12. 

The  value  x  =  15  satisfies  the  enunciation  ;  for,  15  yards 

240 
for  240  cents,  gives     -yr'     or   -1  ^  cents,  for  the  price  of 

J.  o 

one  yard  ;  and  12  yards  for  240  cents,  gives  20.  cents  for  the 
price  of  one  yard,  which  exceeds  16  by  4. 


EQUATIONS  OF  THE   SECOND  DE&REE.   235 

As  to  the  second  solution,  we  can  form  a  new  enuncia- 
tion, with  which  it  will  agree.  For,  going  back  to  the 
equation,  and  changing  x  into  —  ar,  we  have 

240        _  240  240  _   240 

an  equation  which  may  be  considered  the  algebraic  transla- 
tion of  this  problem,  viz. :  A  certain  person  purchased  a  num- 
ber of  yards  of  cloth  for  240  cents  :  if  he  had  paid  the  same 
sum  for  3  yards  more,  it  would  have  cost  him  4  cents  less  per 
yard.     How  many  yards  did  he  purchase  ? 

Ans.  x  =  12,  and  x  =  —  15. 

3.  A  man  bought  a  horse,  which  he  sold  for  24  dollars 
At  this  sale,  he  lost  as  much  per  cent,  upon  the  price  of  his. 
purchase  as  the  horse  cost  him.  What  did  he  pay  for  the 
horse  % 

Let  x  denote  the  number  of  dollars  that  he  paid  for  the 

horse  ;  then,  x  —  24  will  express  the  loss  he  sustained.    But 

x 
as  he  lost  x  per  cent,  by  the  sale,  he  must  have  lost    — — ■ 
r  J  '  100 

upon  each  dollar,  and  upon  x  dollars  he  lost  a  sum  denoted 


whence     x2  —  100.r  =  —  2400  ; 

-v/2500  -  2400  =  50  ±  10. 
x  =  60    and    x  =  40. 

Both  of  these  values  will  satisfy  the  question. 

For,  in  the  first  place,  suppose  the  man  gave  $00  for  the 
horse  and  sold  him  for  24,  he  loses  36.  Again,  from  the 
enunciation,  he    should   lose   60  per  cent,  of   60,  that   is. 


by 

x2  < 

loo'5 

we 

have 

X2 

Too 

=  X 

-24, 

and 

x  = 

50  ± 

Therefore, 

X 

236  ELEMENTARY  ALGEBRA. 

C0    i-  „n      60  X  GO    ,  .  ,     ,  „„    .     . 

-T-zrpr  oi  00,  or  — — — — ,  which  reduces  to  3G  ;  therefore 
100  100 

GO  satisfies  the  enunciation. 

Had  he  paid  $40,  he  would  have  lost  $16  by  the  sale ; 

40 

for,  he  should  lose  40  per  cent,  of  40,  or  40  X  — -^r,    which 

100 

reduces  to  16;  therefore,  40  verifies  the  enunciation. 

4.  A  man  being  asked  his  age,  said  the  square  root  of 
my  own  age  is  half  the  age  of  my  son,  and  the  sum  of  our 
ages  is  80  years  :  what  was  the  age  of  each  1 

Let     x  =     the  age  of  the  father. 
y  =      that  of  the  son. 
Then  by  the  first  condition 

and  by  the  second  condition 

x  +  y  =  80. 
If  we  take  the  first  equation 

V  x  —  o' 
and  square  both  members,  we  have 

y2 

If  we  transpose  y  in  the  second,  we  have 
x  —  80  —  y  : 
from  which  we  find 

y  =  —  2  ±  -/S24  =  1G  ; 

by  taking  the  plus  root,  which  answers  to  the  question  in 
its  arithmetical  sense.  Substituting  this  value,  we  find 
x  =  64.  .         j  Father's  age  G4. 

1G. 


I  Father' 
(  Son's 


EQUATIONS  OF  THE  SECOND  DEGREE.   237 

5.  Find  two  numbers,  such,  that  the  sum  of  their  pro- 
ducts by  the  respective  numbers  a  and  h,  may  be  equal  to 
2s,  and  that  their  product  may  be  equal  to  p. 

Let  x  and  y  denote  the  required  numbers  :  we  then  have 
the  equations 

ax  +  by  — 2s, 

and  xy=p. 

-r.  t>  ~s  —  ax 

From  the  first  y  = . ; 

a 

whence,  by  substituting  in  the  second,  and  reducing, 

ax2  —  2sx  r=  —  bp. 

s     .     1 


Therefore,         x  =  —  ±  —  -y/s1  —  abp, 


a 


and  consequently, 


1 


This  problem  is  susceptible  of  two  direct  solutions,  be- 
cause s  is  evidently  >  ys"2  —  abp ;  but  in  order  that  they 
may  be  real,  it  is  necessary  that  s2>   or  =zabp. 

Let  a  =  b  =  1  ;  the  values  of  x  and  y  reduce  to 


x  =  s  ±  yV2  —  p   and   y  =  s  zp  -y/s2  — p. 

Whence  we  see,  that  the  two  values  of  x  are  equal  to 
those  of  y,  taken  in  an  inverse  order ;  which  shows,  that  if 
s  +  yV2  —  p  represents  the  value  of  x,  s  —  yV2  —  p  will 
represent  the  corresponding  value  of  y,  and  reciprocally. 

This  circumstance  is  accounted  for,  by  observing,  that  In 
this  pai  tic  liar  case,  the  equations  reduce  to 
ix  +  y=2s, 
)         xy—p; 


238  ELEMENTARY     ALGEBRA. 

and  then  the  question  is  reduced  to  finding  two  numbers  of 
which   their  sum   is  2s,  and  their  product  p ;    or   in   other 
words,  to  divide  a  number  2s,  into  two  such  parts,  that  their 
product  may  be  equal  to  a  given  number  p. 
Let  us  now  suppose 

2s  =14    and  p  =  48: 

what  will  then  be  the  values  of  x  and  y  % 

x  =  8  or  0. 


Ans. 

y  =  G  or  8. 

6.  A  grazier  bought  as  many  sheep  as  cost  him  £G0,  and 
after  reserving  fifteen  out  of  the  number,  he  sold  the  re- 
mainder for  £54,  and  gained  2s.  a  head  on  those  he  sold  : 
how  many  did  he  buy  1  Ans.  75. 

7.  A  merchant  bought  cloth  for  which  he  paid  £33*  15s., 
which  he  sold  again  at  £2  8s.  per  piece,  and  gained  by  the 
bargain  as  much  as  one  piece  cost  him:  how  many  pieces 
did  he  buy?  Ans.   15. 

8.  What  number  is  that,  which,  being  divided  by  the  pro- 
duct of  its  digits,  the  quotient  is  3;  and  if  IS  be  added  to 
it,  the  order  of  the  digits  will  be  inverted?  Ans.-  24. 

9.  To  find  a  number,  such  that  if  you  subtract  it  from  10, 
and  multiply  the  remainder  by  the  number  itself,  the  pro- 
duct shall  be  21.  Ans.  7  or  3. 

10.  Two  persons,  A  and  B,  departed  from  different  places 
at  the  same  time,  and  travelled  towards  each  other.  On 
meeting,  it  appeared  that  A  had  travelled  18  miles  more 
than  B  ;  and  that  A  could  have  gone  B's  journey  in  1 5| 
days,  but  B  would  have  been  28  days  in  performing  A's 
journey.     How  fai  did  each  travel  ? 

.  (  A  72  miles. 

B  54  miles. 


EQUATIONS  OF  THE  SECOND  DEGREE.   239 

11.  There  are  two  numbers  whose  difference  is  15,  and 
half  their  product  is  equal  to  the  cube  of  the  lesser  number. 
What  are  those  numbers'?  Ans.  3  and  18. 

12.  What  two  numbers  are  those  whose  sum,  multiplied 
by  the  greater,  is  equal  to  77  ;  and  whoue  difference,  multi- 
plied by  the  lesser,  is  equal  to  12] 

Ans.  4  and  7,  or  §  -y/2  and  y  -y/Z. 

13.  To  divide  100  into  two  such  parts,  that  the  sum  of 
their  square  roots  may  be  14.  Ans.  64  and  36. 

14.  It  is  required  to  divide  the  number  24  into  two  such 
parts,  that  their  product  may  be  equal  to  35  times  their 
difference.  Ans.  10  and  14. 

15.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their 
cubes  is  152.     What  are  the  numbers'?  Ans.  3  and  5. 

16.  Two  merchants  each  sold  the  same  kind  of  stuff; 
the  second  sold  3  yards  more  of  it  than  the  first,  and  to- 
gether they  receive  35  dollars.  The  first  said  to  the  second, 
"  I  would  have  received  24  dollars  for  your  stuff;"  the 
other  replied,  "And  I  should  have  received  12J  dollars  for 
yours.  '     How  many  yards  did  each  of  them  sell? 

.  (1st  merchant  x  =  15  x=  5. 

AUS-     { 2d  «         y=18      °r      y  =  & 

17.  A  widow  possessed  13,000  dollars,  which  she  divided 
into  two  parts,  and  placed  them  at  interest,  in  such  a  man- 
ner, that  the  incomes  from  them  were  equal.  If  she  had 
put  out  the  first  portion  at  the  same  rate  as  the  second,  she 
would  have  drawn  for  this  part  300  dollars  interest ;  and  if 
she  had  placed  the  second  out  at  the  same  rate  as  the  first, 
she  would  have  drawn  for  it  490  dollars  interest.  What 
were  the  two  rates  of  interest  % 

Ans,  7  and  6  per  cent. 


240  ELEMENTARY      ALGEBRA. 


CHAPTER   VII. 

Of  Proportions  and  Progressions. 

135.  Two  quantities  of  the  same  kind  may  be  compaied, 
the  one  with  the  other,  in  two  ways : — 

1st.  By  considering  how  much  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference ;  and,  x 

2d.  By  considering  how  many  times  one  is  greater  or  less 
than  the  other,  which  is  shown  by  their  quotient. 

Thus,  in  comparing  the  numbers  3  and  12  together,  with 
respect  to  their  difference,  we  find  that  12  exceeds  3  by  9  ; 
and  in  comparing  them  together  with  respect  to  their  quo- 
tient, we  find  that  12  contains  3  four  times,  or  that  12  is  4 
times  as  great  as  0. 

The  first  of  these  methods  of  comparison  is  called  Arith- 
metical Proj^ortion,  and  the  second,  Geometrical  Proportion. 

Hence,  Aritlimetical  Proportion  considers  the  relation  of 
quantities  with  respect  to  their  difference,  and  Geometrical 
Proportion  the  relation  of  quantities  with  respect  to  their 
quotient. 


135i  In  how  many  ways  may  two  quantities  be  compared  the  one 
with  the  other  ?  What  does  the  first  method  consider  ?  "What  the 
second  ?  "What  is  the  first  of  these  methods  called  ?  What  is  the 
second  called  ?     How  then  do  you  define  the  two  proportions  ? 


ARITHMETICAL     PROPORTION.  241 

Of  Arithmetical  Proportion  and  Progression. 

136.  If  we  have  four  numbers,  2,  4,  8,  and  10,  of  which 
the  difference  between  the  first  and  second  is  equal  to  the 
difference  between  the  third  and  fourth,  these  numbers  are 
said  to  be  in  arithmetical  proportion.  The  first  term  2  is 
called  an  antecedent,  and  the  second  term  4,  with  which  it  is 
compared,  a  consequent.  The  number  8  is  also  called  ac 
antecedent,  and  the  number  10,  with  which  it  is  compared, 
a  consequent. 

When  the  difference  between  the  first  and  second  is  equal 
to  the  difference  between  the  third  and  fourth,  the  four  num- 
bers are  said  to  be  in  proportion.     Thus,  the  numbers 

2,     4,     8,     10, 

are  in  arithmetical  proportion. 

137.  When  the  difference  between  the  first  antecedent 
and  consequent  is  the  same  as  between  any  two  adjacent 
terms  of  the  proportion,  the  proportion  is  called  an  arith- 
metical progression.  Hence,  a  progression  by  differences,  or 
an  arithmetical  'progression,  is  a  series  in  which  the  succes- 
sive terms  are  continually  increased  or  decreased  by  a  con- 
stant number,  which  is  called  the  common  difference  of  the 
progression. 

Thus,  in  the  two  series 

1,     4,     7,  10,  13,  1G,  19,  22,  25,  .  .  . 
GO.  56,  52,  48,  44,  40,  36,  32,  28,  .  .  . 


13G    When  are  four  numbers  in  arithmetical  proportion  ?     What  is  the 

6rst  called?     What  is  the  second  called?     What  is  the  third  called? 

What  is  the  fourth  called  ? 

11 


242  ELEMENTARY     ALGEBRA. 

the  first  is  called  an  increasing  progression,  of  which  the 
common  difference  is  3,  and  the  second  a  decreasing  pro- 
gression, of  which  the  common  difference  is  4. 

In  general,  let  a,  b,  c,  d,  e,  /*  .  .  .  designate  the  terms  of 
a  progression  by  differences ;  it  has  been  agreed  to  write 
them  thus : 

a.b.c.d.e.f.g.h.i.k... 

This  series  is  read,  a  is  to  5,  as  b  is  to  c,  as  c  is  to  d,  as  d  is 
to  e,  &e.  This  is  a  series  of  continued  equi-diffei-ences,  in 
which  each  term  is  at  the  same  time  an  antecedent  and  a  con- 
sequent, with  the  exception  of  the  first  term,  which  is  only 
an  antecedent,  and  the  last,  which  is  only  a  consequent. 

138.  Let  d  represent  the  common  difference  of  the  pro- 
gression 

a.b.c.e.f.g.h,  &c, 

which  we  will  consider  increasing. 

From  the  definition  of  the  progression,  it  evidently  fol- 
lows that 

b  —  a  +  d,     c=zb+d=a  +  2d,     e  =  c  +  d  =  a  +  Bd; 

and,  in  general,  any  term  of  the  series  is  equal  to  the  Jirsl 
term  plus  as  many  times  the  common  difference  as  there  are 
preceding  terms. 

Thus,  let  I  be  any  term,  and  n  the  number  which  marks 
the  place  of  it :  the  expression  for  this  general  term  is 
I  =  a  +  (n  —  l)d. 


137.  "What  is  an  arithmetical  progression  ?  What  is  the  number  call- 
ed by  -which  the  terms  are  increased  or  diminished  ?  What  is  an  increas- 
ing progression  i  What  is  a  decreasing  progression?  Which  term  ia 
only  an  antecedent  ?     Which  only  a  consequent  ? 


ARITHMETICAL     PROGRESSION.  243 

Hence,  for  finding  the  last  term,  we  have  the  following 

RULE. 

I.  Multiply  the  common  difference  by  the  number  of  terms 
less  one. 

II.  To  the  product  add  the  first  term  :  the  sum  will  be  the 
last  term. 

EXAMPLES. 

The  formula  I  =  a  +  (n  —  l)d  serves  to  find  any  term 
whatever,  without  our  being  obliged  to  determine  all  those 
which  precede  it. 

1.  If  we  make  n  —  1,  we  have  I  =  a  ;  that  is,  the  series 
will  have  but  one  term. 

2.  If  we  make  n  =  2,  we  have  I  =  a  +  d ;  that  is,  the 
series  will  have  two  terms,  and  the  second  term  is  equal  to 
the  first  plus  the  common  difference. 

S.  If  a  =  3  and  d  =  2,  what  is  the  3d  term "?        Ans.  7. 

4.  If  a  =  5  and  d  =  4,  what  is  the  6th  term  %     Ans.  25. 

5.  If  a  =  7  and  d  —  5,  what  is  the  9th  term  ]     Ans.  47. 

6.  If  a  =  8  and  d  =  5,   what  is  the  tenth  term  ? 

J[«s.  53. 

7.  If  a  =  20  and  d  =?  4,  what  is  the  12th  term  1 

Ans.  64. 

8.  If  a  =  40  and  d  =  20,   what  is  the  50th  term  1 

Ans.  1020. 


138.  Give  the  rule  for  finding  the  last  term  of  a  series  when  the  pro- 
greaaiou  is  increasing. 


244  ELEMENTARY     ALGEBRA. 

9.  If  a  =  45  and  d  ~  30,  what  is  the  40th  term  1 

Ans.  1215. 

10.  If  a  =  30  and  d  =  20,  what  is  the  GOth  term  1 

Ans.  1210. 

11.  If  a  =  50  and  d  —  10,  what  is  the  100th  term] 

Ans.  1040. 

1 2.  To  find  the  50th  term  of  the  progression 

1  .  4  .  7  .  10  .  13  .  1G  .  19  .  .  ., 
we  have  I  =  1  +  49  x  3  =  148. 

13.  To  find  the  GOth  term  of  the  progression 

1  .  5  .  9  .  13  .  17  .  21  .  25  .  .  ., 
we  have  I  =  1  +  59  X  4  =  237. 

139.  If  the  progression  were  a  decreasing  one,  we  should 
have 

I  =  a  —  («  —  l)d. 

Hence,  to  find  the  last  term  of  a  decreasing  progression,  we 
have  the  following 


RULE. 

I.  Multiply  the  common  difference  by  the  number  of  terms 
less  one. 

II.  Subtract  the  product  from  the  first  term;  the  remainder 
will  be  the  last  term. 


139i  Give  the  rule  for  finding  the  last  term  of  a  scries,  wheu  the  pro 
greasion  is  decreasing. 


ARITHMETICAL     PROGRESSION.  245 


EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  60,  the 
uumber  of  terms  20,  and  the  common  difference  8  :  what  is 
the  last  term  1 

l=a—{n—V)d    gives    /=60-(20-l)3=G0-57=3. 

2.  The  first  term  is  90,  the  common  difference  4,  and  the 
number  of  terms  15  :  what  is  the  last  term  1  Ans.  34. 

3.  The  first  term  is  100,  the  number  of  terms  40,  and  the 
common  difference  2  :  what  is  the  last  term  %  Ans.  22. 

4.  The  first  term  is  80,  the  number  of  terms  10,  and  the 
common  difference  4  :  what  is  the  last  term  %  Ans.  44. 

5.  The  first  term  is  000,  the  number  of  terms  100,  and 
the  common  difference  5  :  what  is  the  last  term  % 

Ans.  105. 

6.  The  first  term  is  800,  the  number  of  terms  200,  and 
the  common  difference  2  :  what  is  the  last  term  1 

Ans.  402. 

1-10.  A  progression  by  differences  being  given,  it  is  pro- 
posed to  prove  that,  the  sum  of  any  two  terms,  taken  at  equal 
distances  from  the  two  extremes,  is  equal  to  the  sum  of  the  tiro 
extremes. 

That  is,  if  we  have  the  progression 

2  .  4  .  G  .  8  .  10  .  12, 

# 

we  wish  to  prove  generally,  that 

4+10     or     G  +  8, 
is  e^ua)  to  the  sum  of  the  two  extremes  2  and  12. 


246  ELEMENTART      ALGEBRA. 

Let  a.b.c.e./....i.k.lbe  the  proposed 
progression,  and  n  the  number  of  terms. 

We  will  first  observe  that,  if  x  denotes  a  term  which  has 
p  terms  before  it,  and  y  a  term  which  has  p  terms  after  it, 
we  have,  from  what  has  been  said, 

x  =  a  -\-  p  x  d, 
and  y  =  I  —  p  xrf; 

whence,  by  addition,    x  +  y  =  a  +  I, 

which  proves  the  proposition. 

Referring  to  the  previous  example,  if  we  suppose,  in 
the  first  place,  x  to  denote  the  second  term  4,  then  y 
will  denote  the  term  10,  next  to  the  last.  If  x  denotes 
the  3d  term  6,  then  y  will  denote  8,  the  third  term  from 
the  last. 

Having  proved  the  first  part  of  the  proposition,  write  the 
terms  of  the  progression,  as  below,  and  then  again,  in  an 
inverse  order,  viz.  : 

a.b.c.d.e.f...i.k.l. 

I  .  k  .  i c  .  b  .  a. 

Calling  S  the  sum  of  the  terms  of  the  first  progression, 
2S  will  be  the  sum  of  the  terms  of  both  progressions,  and 
we  shall  have 

25  =  (a+0+(H*)+(c+t)...+(»+e)+(*+S)+(?+a). 

4 

Now  since  all  the  parts,  a  +  I,  b  +  k.t  c  +  i  •  ■  .  are  equal 

to  each  other,  and  their  number  equal  to  ?i, 

2S  =  (a  +  0  X  n,     or     S  =  (^\  X  n. 


ARITHMETICAL     PROGRESSION.  247 

Hence,  for  finding  the  sum  of  an  arithmetical   series,  we 
have  the  following 


RULE. 

I.  Add  the  two  extremes  together,  and  take  half  their  sum. 

II.  Multiply  this  half-sum  by  the  number  of  terms;    the 
product  will  be  the  sum  of  the  series. 

EXAMPLES. 

1.  The  extremes  are  2  and  16,  and  the  number  of  terms 
8  :  what  is  the  sum  of  the  series  % 

S+^Xn,    gives    8  =  ^^x8  =  72. 

2.  The  extremes  are  3  and  27,  and  the  number  of  terms 
12  :  what  is  the  sum  of  the  series  1  Ans.  180. 

3.  The  extremes  are  4  and  20,  and  the  number  of  terms 
10  :  what  is  the  sum  of  the  series  1  Ans.  120. 

4.  The  extremes  are  100  and  200,  and  the  number  of 
terms  80  :  what  is  the  sum  of  the  series  1  Ans.  12000. 

5.  The  extremes  are  500  and  GO,  and  the  number  of  terms 
20  :  what  is  the  sum  of  the  series  1  Ans.  5600. 

6.  The  extremes  are  800  and  1200,  and  the  number  of 
terms  50  :  what  is  the  sum  of  the  series  1  Ans.  50000. 


140.  In  every  progression,  what  is  the  sum  of  the  two  extremes 
equal  to  ?  What  is  the  rule  for  finding  the  sum  of  an  arithmetical 
eeries  f 


248  ELEMENTARY     ALGEBRA. 

141.  In  arithmetical  proportion  there  are  five  numbers  to 
be  considered  : — 

1st.  The  first  term,  a. 

2d.    The  common  difference,  d. 

3d.    The  number  of  terms,  n. 

4th.  The  last  term,  I. 

5th.  The  sum,  S. 

The  formulas 

l=a  +  (n  —  l)d  and     S  —  I  — ^-  \  X  n 

contain  five  quantities,  a,  d,  n,  I,  and  S,  and  consequently 
give  rise  to  the  following  general  problem,  viz  :  Any  three 
of  these  Jive  quantities  being  given,  to  determine  the  other 
two. 

"We  already  know  the  value  of  S  in  terms  of  a,  n,  and  I. 

From  the  formula 

I  =  a  +  (n  —  l)d, 
we  find  a  =  I  —  (n  —  \)d. 

That  is  :  The  first  term  of  an  increasing  arithmetical  pro- 
gression is  equal  to  the  last  term,  minus  the  product  of  the 
common  difference  by  the  number  of  terms  less  one. 

From  the  same  formula,  we  also  find 

I  —  a 
d= r. 

n  —  1 

That  is :  In  any  arithmetical  progression,  the  common  differ- 
ence is  equal  to  the  last  term  minus  the  first  term  divided  by 
the  number  of  terms  less  one. 


1 4 1.1  How  many  numbers  are  considered  in  arithmetical  proportion  \ 
What  are  they  ?  In  every  arithmetical  progression,  what  is  the  common 
difference  eaual  to  \ 


ARITHMETICAL     PROGRESSION.  249 

The  last  term  is  1G,  the  first  term  4,  and  the  number  of 
terms  5  :  what  is  the  common  difference  ] 

/  —  a, 

The  formula  d= 

n  —  1 

.     16-4 
give3  a  =  — - — =3. 


2.  The  last  term  is  22,  the  first  term  4,  and  the  number 
of  terms  10  :  what  is  the  common  difference?  Ans.  2. 

142.  The  last  principle  affords  a  solution  to  the  following 
question  : 

To  find  a  number  ra  of  arithmetical  means  between  two 
given  numbers  a  and  b. 

To  resolve  this  question,  it  is  first  necessary  to  find  the 
common  difference.  Now,  we  may  regard  a  as  the  first 
term  of  an  arithmetical  progression,  b  as  the  last  term,  and 
the  required  means  as  intermediate  terms.  The  number  of 
terms  of  this  progression  will  be  expressed  by  m  +  2. 

Now,  by  substituting  in  the  above  formula,  b  for  I,  and 
m  4   2  for  n,  it  becomes 

b  —  a  b  —  a 

~ m+ 2  -  1 = m+  1  ' 

that  is  :  The  common  difference  of  the  required  progression  is 
obtained  by  dividing  the  difference  between  the  given  numbers 
a  and  b,  bg  the  required  number  of  means  plus  one. 


142.  How  do  you  find  any  number  of  arithmetical  means  between 
iwo  given  numbers  ? 

II* 


250  ELEMENTARY      ALGEBRA. 

Having  obtained  the  common  difference,  form  the  second 
term  of  the  progression,  or  the  first  arithmetical  mean,  by 
adding  d  to  the  first  term  a.  The  second  mean  is  obtained 
by  augmenting  the  first  mean  by  <f,  &c. 

1.  Find  three  arithmetical  means  between  the  extremes 
2  and  ]  8. 

The  formula  c?  = 


m  +  1 


,7        18~2        A 

gives  d  =  — - —  =  4; 


hence,  the  progression  is 

2  .  C  .  10  .  14  .   18. 

2.  Find  twelve  arithmetical  means  between  12  and  77. 
b  —  a 


The  formula  d  = 


m  +  1 


7      77-12      E 
gives  d  = — — =5. 

lo 
Hence,  the  progression  is 

12  .   17  .  22  .  27  ....  77. 

143.  Remark. — If  the  same  number  of  arithmetical 
means  are  inserted  between  all  the  terms,  taken  two  and 
two,  these  terms,  and  the  arithmetical  means  united,  will 
form  but  one  and  the  same  progression. 

For,  let  a  .  b  .  c  .  e  .  /  .  .  .  be  the  proposed  pro- 
gression, and  m  the  number  of  means  to  be  inserted  be- 
tween a  and  6,  b  and  c,  c  and  e  .  .  .  .    &c. 


ARITHMETICAL     PROGRESSION.  251 

From  what  has  just  been  said,  the  common  difference  ot 
each  partial  progression  will  be  expressed  by 


m  +  1'      m  -j-  1'       m  +  1 

expressions  which  are  equal  to  each  other,  since  a,  b,  c  .  .  . 
are  in  progression  :  therefore,  the  common  difference  is  the 
same  in  each  of  the  partial  progressions ;  and  since  the 
last  term  of  the  first,  forms  the  first  term  of  the  second,  &c, 
we  may  conclude  that  all  of  these  partial  progressions  form 
a  single  progression. 


EXAMPLES. 

1.  Find  the  sum  of  the  first  fifty  terms  of  the  progres- 
sion  2.9.   1G  .  23  .  .  . 

For  the  50th  term  we  have 

1—2  +  49  X  7  =  345. 


50 
Hence,    S  =  (2  +  345)  X  ^r  —  347  X  25  =  8G75. 

fit 


2.  Find  the  100th  term  of  the  series  2  .  9  .  16  .  23  .  .  . 

Ans.  G95. 

3.  Find  the  sum  of  100  terms  of  the  series  1.3.5. 
7.9...  Am.  10000. 

4.  The  greatest  term  is  70,  the  common  difference  3,  and 
the  number  of  terms  21  :  what  is  the  least  term  and  the 
sum  of  the  series  ? 

Ans.  Least  term  10 ;  sum  of  series  840, 


252  ELEMENTARY     ALGEBRA. 

5.  The  first  term  is  4,  the  common  difference  8,  and  the 
number  of  terms  8  :  what  is  the  last  term,  and  the  sum  of 
the  series  ] 

Last  term  60. 


Ans. 

Sum     =  2oo. 

6.  The  first  term  is  2,  the  last  term  20,  and  the  number 
of  terms  10  :  what  is  the  common  difference  1 

Ans.  2. 

7.  Insert  four  means  between  the  two  numbers  4  and  19  : 
what  is  the  series  1 

Ans.  4  .  7  .   10  :   13  .   1G  .   19. 

8.  The  first  term  of  a  decreasing  arithmetical  progression 
is  10,  the  common  difference  one-third,  and  the  number  of 
terms  21  :  required  the  sum  of  the  series. 

Ans.  140. 

9.  In  a  progression  by  differences,  having  given  the  com- 
mon difference  6,  the  last  term  185,  and  the  sum  of  the 
terms  2945  :  find  the  first  term,  and  the  number  of  terms. 

Ans.  First  term  =  5  ;  number  of  terms  31. 

10.  Find  nine  arithmetical  means  between  each  antece- 
dent and  consequent  of  the  progression  2.5.8.11.14... 

Ans.  Common  difi,  or  d  =.  0.3. 

11.  Find  the  number  of  men  contained  in  a  triangular 
battalion,  the  first  rank  containing  one  man,  the  second  2, 
the  third  3,  and  so  on  to  the  n'\  which  contains  n.  In  other 
words,  find  the  expression  for  the  sum  of  the  natural  num- 
bers 1,  2,  3  .  .  .,  from  1  to  n  inclusively. 

Ans/S=:7^±V 


GEOMETRICAL     PROPORTION.  253 

12.  Find  the  sum  of  the  n  first  terms  of  the  progression 
of  uneven  numbers  1,  3,  5,  7,  9  .  .  .  Aas.  S  =  w2. 

13.  One  hundred  stones  being  placed  on  the  ground  in  a 
straight  line,  at  the  distance  of  2  yards  apart,  how  far  will  a 
person  travel  who  shall  bring  them  one  by  one  to  a  basket, 
placed  at  a  distance  of  2  yards  from  the  first  stone  1 

Ans.  11  miles,  840  yards. 

Geometrical  Proportion  and  Progression. 

144.  Ratio  is  the  quotient  arising  from  dividing  one 
quantity  by  another  quantity  of  the  same  kind.  Thus,  if 
the  numbers  3  and  6  have  the  same  unit,  the  ratio  of  3  to  6 
will  be  expressed  by 

*-* 

And  in  general,  if  A  and  B  represent  quantities  of  the  same 
kind,  the  ratio  of  A  to  B  will  be  expressed  by 

B_ 

A' 

145.  If  there  be  four  numbers 

2,     4,     8,     16, 

having  such  values  that  the  second  divided  by  the  first  is 
*qual  to  the  fourth  divided  by  the  third,  the  numbers  are 

144.  What  is  ratio  ?     What  is  the  ratio  of  3  to  6  ?     Of  4  to  12  t 


254  ELEMENTARY    ALGEBRA. 

said  to  be  in  proportion.  And  in  general,  if  there  be  four 
quantities,  A,  B,  C,  and  I),  having  such  values  that 

B__D_ 

~A~~C' 

then  A  is  said  to  have  the  same  ratio  to  B  that  C  has  to  D, 
or,  the  ratio  of  A  to  B  is  equal  to  the  ratio  of  C  to  D. 
When  four  quantities  have  this  relation  to  each  other,  com- 
pared together  two  and  two,  they  are  said  to  be  in  geomet- 
rical proportion. 

To  express  that  the  ratio  of  A  to  B  is  equal  to  the  ratio 
of  C  to  Z>,  we  write  the  quantities  thus  : 

A  :  B  ::   0  :  D ; 

and  read,  A  is  to  B  as  C  to  D. 

The  quantities  which  are  compared,  the  one  with  the 
other,  are  called  terms  of  the  proportion.  The  first  and  last 
terms  are  called  the  two  extremes,  and  the  second  and  third 
terms,  the  two  means.  Thus,  A  and  D  are  the  extremes, 
and  B  and  C  the  means. 

146.  Of  four  proportional  quantities,  the  first  and  third 
are  called  the  antecedents,  and  the  second  and  fourth  the 
consequents  ;  and  the  last  is  said  to  be  a  fourth  proportional 
to  the  other  three,  taken  in  order.  Thus,  in  the  last  pro- 
portion A  and  C  are  the  antecedents,  and  B  and  D  the  con- 
sequents. 

145.  What  is  proportion?  How  do  you  express  that  four  numbeis 
are  in  proportion  ?  What  are  the  numbers  called  ?  What  are  the  first 
and  fourth  terms  called?     What  the  second  and  third? 

14G.  In  four  proportional  quantities,  what  are  the  first  and  third  called  t 
What  the  second  and  fourth  8 


GEOMETRICAL     PROPORTION.  255 

147.  Three  quantities  are  in  proportion  when  the  first  has 
the  same  ratio  to  the  second  that  the  second  has  to  the 
third  ;  and  then  the  middle  term  is  said  to  be  a  mean  pro- 
portional between  the  other  two.     For  example, 

3  :  6  :  :  6  :   12 ; 

and  G  is  a  mean  proportional  between  3  and  12. 

148.  Quantities  are  said  to  be  in  proportion  by  inversion, 
or  inversely,  when  the  consequents  are  made  the  antecedents 
and  the  antecedents  the  consequents. 

Thus,  if  we  have  the  proportion 

3  :  G  :  :  8  :   16, 
the  inverse  proportion  would  be 

6  :  3  :  :   1G  :  8. 

149.  Quantities  are  said  to  be  in  proportion  by  alterna- 
tion, or  alternately,  when  antecedent  is  compared  with  ante 
cedent  and  consequent  with  consequent. 

Thus,  if  we  have  the  proportion 

3  :  6  :  :  8  :  16, 
the  alternate  proportion  would  be 

3  :  8  :  :  6  :  16. 


147.  When  are  three  quantities  proportional?     What  is  the  middle 
one  called  ?    - 

148.  When  are  quantities  said  to  be  in  proportion  by  inversion,  or  hi 
vcrsely  ? 

149.  When  are  quantities  in  proportion  by  alternation  ? 


256  ELEMENTARY    ALGEBRA. 

150.  Quantities  are  said  to  be  in  proportion  by  composi- 
tion, when  the  sum  of  the  antecedent  and  consequent  is 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

2  :  4  :  :  8  :   16, 
the  proportion  by  composition  would  be 

2  +  4  :  4  :  :  8  +  16  :   16; 
that  is,  6:4::  24  :   16. 

151.  Quantities  are  said  to  be  in  proportion  by  division, 
when  the  difference  of  the  antecedent  and  consequent  is 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

3  :  9  :  :  12  :  36, 

the  proportion  by  division  will  be 

9  -  3  :  9  :  :  36  -  12  :  36 ; 
that  is,  6  :  9  :  :  24  :  36. 

152.  Equi-multiples  of  two  or  more  quantities  are  the 
products  which  arise  from  multiplying  the  quantities  by  the 
same  number. 

Thus,  if  we  have  any  two  numbers,  as  6  and  5,  and  mul- 
tiply them  both  by  any  number,  as  9,  the  equi-multiples 
will  be  54  and  45  ;  for 

6  X  9  —  54,     and     5  x  9  =  45. 

150    "When  are  quantities  in  proportion  by  composition  ? 

151.  When  are  quantities  in  proportion  by  division  ?    - 

152.  "What  are  equi-multiples  of  two  or  more  quantities  ? 


GEOMETRICAL     PROPORTION.  257 

Also  m  X  A  and  m  x  B  are  equi-multiples  of  A  and  B,  the 
common  multiplier  being  m. 

153.  Two  quantities  A  and  i?,  which  may  change  their 
values,  are  reciprocally  or  inversely  proportional,  when  one  is 
proportional  to  unity  divided  by  the  other,  and  then  their 
product  remains  constant. 

We  express  this  reciprocal  or  inverse  relation  thus  : 

,         1 
Ac°B 

in  which  A  is  said  to  be  inversely  proportional  to  B. 

154.  If  we  have  the  proportion 

A  :  B  :  :   C  :  D% 

E       D 

we  have,  —  =  — ,    (Art.  145) ; 

and  by  clearing  the  equation  of  fractions,  we  have 

BC  —  AD. 

That  is,  Of  four  proportional  quantities,  the  product  of  the 
two  extremes  is  equal  to  the  product  of  the  two  means. 

This  general  principle  is  apparent  in  the  proportion  be- 
tween the  numbers 

2  :  10  :  :  12  :  60, 
which  gives         2  X  60  =  10  X  12  =  120. 


153.  When  are  two  quantities  said  to  be  reciprocally  proportional? 

154.  If  four  quantities  are  proportional,  •what  is  the  product  of  the 
two  means  equal  to  ? 

12 


258  ELEMENTARY     ALGEBRA. 

155.  If  four  quantities,  A,  B,  C,  D,  are  so  related  to  each 
other,  that 

AxD  =  Bx  0, 

we  shall  also  have  —r  =  -=-  \ 

A        0 

and  hence,  A  :  B  : :   0  :  D. 

That  is  :  If  the  product  of  two  quantities  is  equal  to  the  pro- 
duct of  two  other  quantities,  two  of  them  may  be  made  the  ex- 
tremes, and  the  other  two  the  means  of  a  proportion. 
Thus,  if  we  have 

2X0-4x4, 
we  also  have 

2  :  4  :  :  4  :  8. 

156.  If  we  have  three  proportional  quantities 

A  :  B  :  :  B  :    0, 

B        C 
we  have,  -  ^  --, 

hence,  B*  -  AC. 

That  is :  If  three  quantities  are  proportional,  the  square  of 
the  middle  term  is  equal  to  the  product  of  the  two  extremes. 
Thus,  if  we  have  the  proportion 

3  :  G  :  :  6  :   12, 
we  shall  also  have 

6  X  6  =  62  =  3  X  12  =  30. 

155i  If  the  product  of  two  quantities  is  equal  to  (he  product  of  two 
other  quantities,  may  the  four  be  placed  in  a  proportion  ?     Hrw  ? 

156i  If  three  quantities  are  proportional,  what  is  the  product  of  the 
extremes  equal  to  ? 


GEOMETRICAL  PROPORTION.        259 

157.     If  we  have 

7?         7) 

A  i  B  :  :   C  :  D,  and  consequently   —  —  —■> 

(j 
multiply  both  members  of  the  last  equation  by  — ,    and 

we  then  obtain, 

C  _D 
A~B' 

and  hence,  A  :   C  :  :  B  :  D. 

That  is:  If  four  quantities  are  proportional,  they  will  be  in 
proportion  by  alternation. 

Let  us  take,  as  an  example, 

10  :  15  :  :  20  :  30. 
We  shall  have,  by  alternating  the  terms, 

10  :  20  :  :  15  :  30. 
15S.  If  we  have 

A  :  B  :  :   C  :  D    and     A  •'  B  :  :  E  :  F, 
we  shall  also  have 

B       I)        ,    B       F 

—  =  77    and    —  =  —  ; 
AC  Ah 

I)       F 
hence,  — -  =  —    and     G  :  D  :  :  E  ;  F. 

O       E 

That  is :    If  there  are  two  sets  of  proportions  having  an 


157.  If  four  quantities  are  proportional,  will  they  be  in  proportion  by 
alternation  ? 


260  ELEMENTARY     ALGEBRA. 

antecedent  and  consequent  in  the  one  eqval  to  an  antecedent 
and  consequent  of  the  other,  the  remaining  terms  will  be  pro- 
portional. 

If  we  have  the  two  proportions 

2  :  6  :  :  8  :  24     and     2  :  G  :  :  10  :  30, 
we  shall  also  have 

8  :  24  :  :  10  :  30. 

159.  If  we  have 

H        D 

A  :  B  :  :    C  :  D,    and  consequently.    —  =  — , 

A         C 

we  have,  by  dividing  1  by  each  member  of  the  equation 

A        C 

jr  =  —  ,  and  consequently    B  :  A  :  :  D  x   C. 

That  is :  Four  proportional  quantities  will  be  in  proportion^ 
when  taken  inversely. 

To  give  an  example  in  numbers,  take  the  proportion 
7  :  14  :  :  8  :  16; 
then,  the  inverse  proportion  will  be 

14  :  7  :  :  1G  :  8, 
in  which  the  ratio  is  one-half. 

160.  The  proportion 

A  :  B  :  :   C  :  D     gives     AxD—Bx  C. 


15S,  If  you  have  two  sets  of  proportions  having  an  antecedent  and 
consequent  in  each,  equal;  what  "will  follow  ? 

159.  Ij  four  quantities  are  in  proportion,  will  they  be  in  proportion 
when  taken  inversely  ? 


GEOMETRICAL  PROPORTION.       2G1 

To  each  member  of  the  last  equation  add  B  X  D.     We 
shall  then  have 

(.1+  B)  xD={C+B)x  B; 

and  by  separating  the  factors,  we  obtain 

A  +  B  :  B  :  :   C  +  D  :  D. 

If,  instead  of  adding,  we  subtract  B  x  D  from  both  mem- 
bers, we  have 

(A-B)  xD={C-D)  xB; 

which  gives 

A  -  B  :  B  :  :   C  -  D  :  D. 

That  is  :  If  four  quantities  are  proportional,  they  will  be  in 
proportion  by  composition  or  division. 

Thus,  if  we  have  the  proportion 

9  :  27   :  :   16  :  48, 
we  shall  have,  by  composition, 

9  +  27  :  27  :  :   1G  +  48  :  48 ; 
that  is,  36  :  27  :  :  G4  :  48, 

in  which  the  ratio  is  three-fourths. 

The  same  proportion  gives  us,  by  division, 
27-9  :  27  ::  48-16  :  48; 
that  is,  18  :  27  :  :  32  :  48, 

in  which  the  ratio  is  one  and  one-half. 


1G0.  If  four  quantities  are  in  proportion,  will  they  bo  in  proportion 
bv  composition  ?  Will  they  be  in  proportion  by  division  i  What  is  the 
difference  between  composition  and  division  ? 


262  ELEMENTARY     ALGEBRA. 

161.  If  "we  have 

B___  D_ 

and  multiply  the  numerator  and  denominator  of  the  first 
member  by  any  number  m,  we  obtain 

— :  =  — -  and   mA  :  mB  :   :    C  :  D. 
mA       0 

That  is :  Equal  multiples  of  two  quantities  have  lite  same 
ratio  as  the  quantities  themselves. 

For  example,  if  we  have  the  proportion 

5  :   10  :  :   12  :  24, 

and  multiply  the  first  antecedent  and  consequent  by  G,  we 
have 

30  :  GO  :  :  12  :  24, 

in  which  the  ratio  is  still  2. 

162.  The  proportions 

A  :  B  :  :   C  :  D  and  A  :  B  :  :  E  :  F, 

give         A  X  D  =  B  x  C  and    A  X  F  =  B  x  E; 

adding  and  subtracting  these  equations,  we  obtain 

A(D±F)  =B(C±E),  or  A  :  B   :  :    C±E  :  B±F. 

That  is :  If  C  and  D,  the  antecedent  and  consequent,  be  aug- 
mented or  diminished  by  quantities  E  and  F,  which  have  the 
same  ratio  as  C  to  D,  the  resulting  Quantities  will  also  have 
the  same  ratio. 

161.  Have  equal  multiples  of  two  quantities  the  same  ratio  as  the 
quantities  3 

162.  Suppose  the  antecedent  aud  consequent  be  augmented  or  diaiiD 
Ished  by  quantities  having  the  same  ratio  ? 


GEOMETRICAL  PROPORTION.        263 

Let  us  take,  as  an  example,  the  proportion 
9  :  18  :  :  20  :  40, 
in  which  the  ratio  is  2. 

If  we  augment  the  antecedent  and  consequent  by  the 
numbers  15  and  30,  which  have  the  same  ratio,  we  shall 
have 

9  +  15  :   18  +  SO  :  :  20  :  40 ; 
that  is,  24  :  48  :  :  20  :  40, 

in  which  the  ratio  is  still  2. 

If  we  diminish  the  second  antecedent  and  consequent  by 
these  numbers  respectively,  we  have 

9  :  18  :  :  20  -  15  :  40  —  30 ; 
that  is,  9  :  18  :  :  5  :  10, 

in  which  the  ratio  is  still  2. 

163.  If  we  have  several  proportions 
A  :  B  :  :   C  :  Z>,   which  gives   A  x  D  =  £  x  C, 
A  :  B  :  :  E  :  F,        "        "       A  X  F  =  B  x  E, 
A  :  B  :  :   G  :  IT,        "        "       Ax  H =  B  X  G, 

we  shall  have,  by  addition, 

A{D  +  F+  H)  =B(C  +  E+  60; 

and  by  separating  the  factors, 

A  :  B  :  :   C+E+  G  :  D+  F+  H. 

That  is  :  In  any  number  of  proportions  having  the  tame 
ritio,  any  antecedent  will  be  to  its  consequent,  as  the  sum  of 
the  antecedents  to  the  sum  of  the  consequents. 


264 


ELEMENTARY     ALGEBRA, 


Let  us  take,  for  example, 

2  :  4  :  :  G  :   12     and     1   :  2  :  :  3  :  C,     &c 
Then  2  :  4  :  :  6  +  3  :  12  +  6; 

that  is,  2  :  4  :  :  9  :   18, 

in  which  the  ratio  is  still  2. 

164.  If  we  have  four  proportional  quantities 

7?         T) 
A  :  B  :  :    C  :  Z>,    we  have     —=-—-; 

and  raising  both  members  to  any  power  whose  exponent  is 
w,  or  extracting  any  root  whose  index  is  n,  we  have, 

B"       D» 

—  =  — ,     and  consequently 

A*       C'"'  ^         J 

That  is  :  If  four  quantities  are  proportional,  their  like  powers 
or  roots  will  be  proportional. 
If  we  have,  for  example, 


2 

.  4 

:  3 

:  6, 

we  shall  have 

22 

42 

:  32 

:  62- 

that  is, 

4 

16 

:  9 

:  36, 

in  which  the  terms  are  proportional,  the  ratio  being  4. 
165.  Let  there  be  two  sets  of  proportion's, 

A  :  B 


C  :  D      which  gives     —  =  — , 


E  :  F 


Q  :  R, 


F_ 

E 


a' 


163.  In  any  number  of  proportions  having  the  same  ratio,  how  fril) 
any  one  antecedent  be  to  its  consequent  ? 

164.  In  four  proportional  quantities,  how  are  like  powers  or  roots? 


OE0MET1UCAI,     PROGRESSION.  265 

Multiply  them  together,  member  by  member,  we  have 
??~  =  ~^.     v>h,ch  gives     AE  :  BF  :  :    CG  :  DR. 

AJii  C(x 

That  is  :  In  two  sets  of  proportional  quantities,  the  products 
of  the  correspond'mg  terms  are  p-opc-rtional. 

Thus,  if  we  liavc  the  two  proportions 
8  :   16  :  :   10  :     20 
and  3  :     4  :  :     6  :       8, 

we  shall  have  24  :  64  :  :  00  :  160. 

Geometrical  Progresssion. 

166.  We  have  thus  far  only  considered  the  case  in  which 
the  ratio  of  the  first  term  to  the  second  is  the  same  as  that 
of  the  third  to  the  fourth. 

If  we  have  the  farther  condition,  that  the  ratio  of  the 
second  term  to  the  third  shall  also  be  the  same  as  that  of 
the  first  to  the  second,  or  of  the  third  to  the  fourth,  we  shall 
have  a  series  of  numbers,  each  one  of  which,  divided  by  the 
preceding  one,  will  give  the  same  ratio.  Hence,  if  any 
term  be  multiplied  by  this  quotient,  the  product  will  be  the 
succeeding  term.  A  series  of  numbers  so  formed  is  called 
a  geometrical  progression.     Hence, 

A  Geometrical  Progression,  or  progression  by  quotients,  is  a 
series  of  terms,  each  of  which  is  equal  to  the  preceding  term 


1G5i  In  two  sets  of  proportions,  how  are  the  products  of  the  corre»« 
ponding  terms? 

12 


266  ELEMENTARY     ALGEBRA. 

multiplied  by  a  constant  number,  which  number  is  called  the 
ratio  of  the  progression.     Thus, 

1   :  3  :  9  :  27  :  81   :  243,  &c, 

is  a  geometrical  progression,  in  which  the  ratio  is  3.  It  i» 
written  by  merely  placing  two  dots  between  the  terms. 

AJ>o,  G4  :  32  :   16  :  8  :  4  :  2  :   1 

is  a  geometrical  progression,  in  which  the  ratio  is  one-half. 

In  the  first  progression  each  term  is  contained  three  times 
in  the  one  that  follows,  and  hence  the  ratio  is  3.  In  the 
second,  each  term  is  contained  one-half  times  in  the  one 
which  follows,  and  hence  the  ratio  is  one-half. 

The  first  is  called  an  increasing  progression,  and  the 
second  a  decreasing  progression. 

Let  a,  b,  c,  d,  e,  /,  .  .  .  be  numbers  in  a  progression  by 
quotients  ;  they  are  written  thus  : 

a  :  b  :  c  :  d  :  e  :  f  :  g  .  .  . 

and  it  is  enunciated  in  the  same  manner  as  a  progression  by 
differences.  It  is  necessary,  however,  to  make  the  distinc- 
tion, that  one  is  a  series  formed  by  equal  differences,  and 
the  other  a  series  formed  by  equal  quotients  or  ratios.  It 
should  be  remarked  that  each  term  is  at  the  same  time  an 
antecedent  and  a  consequent,  except  the  first,  which  is  only 
an  antecedent,  and  the  last,  which  is  only  a  consequent. 


1GG.  "What  is  a  geometrical  progression  ?  "What  is  the  ratio  of  the 
progression  ?  If  any  term  of  a  progression  be  multiplied  by  the  ratio 
what  will  the  product  be?  If  any  term  be  divided  by  the  ratio,  what 
will  the  quotient  be?  How  is  a  progression  by  quotients  written! 
Which  of  the  terms  is  only  an  antecedent  ?  Which  only,  a  consequent/ 
How  may  each  of  the  others  be  considered? 


GEOMETRICAL     PROGRESSION.  267 

]  67.  Let  q  denote  the  ratio  of  the  progression 

a  :  b  :  c  :   d  .  .  .  ; 

q  being  >1  when  the  progression  is  increasing,  and  g<  1 
when  it  is  decreasing.     Then,  since 

b  c  d  e 

we  have 

b  =  ay,     c  =  bq  =  aq2,     d  — ■  cq  =  aq3,     e  =  dq  =  aq*t 
f=eq  =  aq5  .  .   .  ; 

that  is,  the  second  term  is  equal  to  aq,  the  third  to  aq2,  the 
fourth  to  aq3,  the  fifth  to  aq4,  &c.  ;  and  in  general,  the  nth 
term,  that  is,  one  which  has  n  —  1  terms  before  it,  is  ex- 
pressed by   aq"^1. 

Let  I  be  this  term ;  we  then  have  the  formula 

I  =  aq"^~ 1, 

by  means  of  which  we  can  obtain  any  term  without  being 
obliged  to  find  all  the  terms  which  precede  it.  Hence,  to 
find  the  last  term  of  a  progression,  we  have  the  following 

RULE. 

I.  liaise  the  ratio  to  a  power  whose  exponent  is  one  less  than 
the  number  of  terms. 

II.  Multiply  the  power  thus  found  by  the  first  term :  the 
product  will  be  the  required  term. 

167.  By  what  letter  do  we  denote  the  ratio  of  a  progression  f  In  an 
increasing  progression  is  q  greater  or  less  than  1  ?  In  a  decreasing  pro- 
gression is  q  greater  or  less  than  1  ?  If  a  is  the  first  term  and  q  the 
ratio,  what  is  the  second  *erm  equal  to  ?  What  the  third  ?  What  the 
fourth  ?  What  is  the  laat  term  equal  to  ?  Give  the  rule  for  finding  the 
ast  term. 


268  ELEMENTARY      ALGEBRA. 

EXAMPLES. 

1.  Find  the  5th  term  of  the  progression 

2  :  4  :  8  :   16  .  . 
in  which  the  first  term  is  2  and  the  common  ratio  2, 
5th  term  =  2  x  24  =  2  x  16  =  32     Ans. 

2.  Find  the  8th  term  of  the  progression 

2  :  6  :  18  :  54  .  .  . 

8th  term  =  2  X  37  =  2  x  2187  =  4374     Anr 

3.  Find  the  6th  term  of  the  progression 

2  :  8  :  32  :   128  .   .  . 
6th  term  =  2  x  45  —  2  X  1024  —  2048     Ant 

4.  Find  the  7th  term  of  the  progression 

3  :  9  :  27  :  81   .  .  . 

7th  term  =  3  X  35  =  3  x  729  =  2187     Ans 

5.  Find  the  6th  term  of  the  progression 

4  :   12  :  36  :   108  .  .   . 
6th  term  =  4  X  35  =  4  x  243  =  972     Ans. 

6.  A  person  agreed  to  pay  his  servant  1  cent  for  the  firs 
day,  two  for  the  second,  and  four  for  the  third,  doubling 
every  day  for  ten  days:  how  much  did  he  receive  on  the 
tenth  day?  Ans.  $5.12. 


GEOMETRICAL     PROGRESSION.  269 

7.  What  is  the  8th  term  of  the  progression 

9  :  3G  :  144  :  576  .  .  . 
8th  term  =  9  x  47  =  9  X  16384  =  147456     Ans. 

8.  Find  the  12th  term  of  the  progression 

64  :  16  :  4  :  1  :  4"  •  •  • 
4 

/  i  \  n        43         J  1 

12th  terra  =  64  (  —  )     =  -rrr  =  ^i  =  ^^     Ans' 
\4/  4n        48      65536 

168.  We  will  now  proceed  to  determine  the  sum  of  n 
terms  of  a  progression 

a  :  b  :  c  :  d  :  e  :  f  :  .  .  .  :  i  :  k  :  I; 

I  denoting  the  nth.  term. 

We  have  the  equations  (Art.  167), 

b  =  aq,     c  =  bq,     d  =  cq,     e  =  dq,  .  .   .  k=iq,     l=z  kq  , 

and  by  adding  them  all  together,  member  to  member,  we 
deduce 

Sum  0/  1st  members.  Sum  of  2d  members, 

b  +  c  +  d+e+  .  .  .   +k+k=z(a  +  b  +  c  +  d+  .  .  .  +i+k)q; 

in  which  we  see  that  the  first  member  contains  all  the  terms 
but  a,  and  the  polynomial  within  the  parenthesis  in  the 
second  member  contains  all  the  terms  but  I.  Hence,  if  we 
call  tlv>  sum  of  the  terms  S,  we  have 

S-     a  =  {S-l)q  =  Sq-lq,    c    Sq-S  =  lq-a; 

whence  S  =  — — . 

q—  1 


270  ELEMENTARY     ALGEBRA. 

Therefore,  to  obtain  the  sum  of  all  the  terms  or  sum  of  the 
series  of  a  geometrical  progression,  we  have  the 

RULE. 

I.  Multiply  the  last  term  by  the  ratio. 

IJ,  Subtract  the  first  term  from,  the  product. 

111.  Divide  the  remainder  by  the  ratio  diminished  by  unity 
and  the  quotient  will  be  the  sum  of  the  series. 

1.  Find  the  sum  of  eight  terms  of  the  progression 

2  :  6  :   18  :  54  :   1G2  .  .  .  2  X  37  =  4374. 

q  —  1  2 

2.  Find  the  sum  of  the  progression 

2  :  4  :  8  :  16  :  32. 

q-\  1 

3.  Find  the  sum  of  ten  terms  of  the  progression 

2  :  G  :  18  :  54  :  162  .  .  .  2  X  39  =  39366. 

Ans.  59048 

4.  What  debt  may  be  discharged  in  a  year,  or  twelve 
months  by  paying  $1  the  first  month,  $2  the  second  month, 


163.   Give  the  rule  for  finding  the  sum  of  tbe  series.     "What  is  the  first 
dtep  »     What  tho  secor.d  ?     "What  the  third  ? 


GEOMETRICAL     PROGRESSION.  271 

$4   the  third  month,  and  so  on,  each  succeeding   payment 
being  double  the  last ;  and  what  will  be  the  last  payment  ] 

j  Debt.     .     .       $4095. 

(  Last  payment,  $204s. 

5.  A  gentleman  married  his  daughter  on  New  Year's  day, 
and  gave  her  husband  Is.  towards  her  portion,  and  was  to 
double  it  on  the  first  day  of  every  month  during  the  year : 
what  was  her  portion  1  Ans.  £204   15s. 

6.  A  man  bought  10  bushels  of  wheat  on  the  condition 

that  he  shuuld  pay  1  cent  for  the  first  bushel,  3  for  the  second, 

9  for  the  third,  and  so  on  to  the  last :  what  did  he  pay  for 

the  last  bushel  and  for  the  ten  bushels  ? 

\  Last  bushel,  1 190,83. 
Ans. 


'•{ 


Total  cost,  $295,24. 
7.  A  man  plants  4  bushels  of  barley,  which,  at  the  first 
harvest,  produced  32  bushels  ;  these  he  also  plants,  which, 
in  like  manner,  produce  8  fold;  he  again  plants  all  his  crop, 
and  again  gets  8  fold,  and  so  on  for  10  years  :  what  is  his 
last  crop,  and  what  the  sum  of  the  series  1 

Last,  1407374883553285i* 


'  Sum, 160842843834GG0. 
169.  When    the    progression    is    decreasing,    we    have 
q  <  1  and  /  <  a ;  the  above  formula 

Iq  —  a 

for  the  sum  is  then  written  under  the  form 

a  —  lq 

in  order  that  the  two  terms  of  the  fraction  may  be  positive. 


ll!9.   What  is  the  formula  for  the  sum  of  the  series  of  a  decreasing 
progression  ? 


272  ELEMENTARY    ALGEBRA. 

1.  Tind  the  sum  of  the  terms  of  the  progression 

32  :  1G  :  8  :  4  :  2 

S==^=A= *=ol 

1  —   q  1  1 

2"  2" 

2.  Find  the  sum  of  the  first  twelve  terms  of  the  progression 

(i)" 


1  / 1  V1  1 

64  :  16  :  4  :  1  :  —  :...:  64 


4  \4/   '  65536 


r^rL^X-i 


„_«— fy_  05536      4  _  65536_  65535 

r=^-  £  ~  3~  ~     +  19660b 

4 

170. — Remark.  We  perceive  that  the  principal  difficulty 
consists  in  obtaining  the  numerical  value  of  the  last  term,  a 
tedious  operation,  even  when  the  number  of  terms  is  not 
very  great. 

3.  find  the  sum  of  6  terms  of  the  progression 

512  :  128  :  32  .  .  . 

Am.  682|. 

4.  Find  the  sum  of  seven  terms  of  the  progression 

2187  :  729  :  243  .  .  . 

Ans.  3279. 

5.  Find  the  sum  of  six  terms  of  the  progression 

972  :  324  :  108  .  .  . 

Ans.  1456. 

6.  Find  the  sum  of  8  terms  of  the  progression 

147456  :  3G864  :  9216  ... 

Am.  196605. 


GEOMETRICAL     PROGRESSION.  273 

Of  Progressions  having  an  infinite  number  of  terms. 
171.  Let  there  be  the  decreasing  progression 
a  :  b  :  c  :  d  :  e  :  f  :  .  .  . 
containing  an  indefinite  number  of  terms.     In  the  formula 

\-q 

substitute  for  /  its  value  ag-"-1  (Art.  167),  and  we  have 

g  _  a  -  aqn 
l'  —  q* 

which  expresses   the  sum  of  n   terms  of  the  progression. 
This  may  be  put  under  the  form 


S 


a  aq" 


1-q        l-q 

Now,  since  the  progression  is  decreasing,  q  is  a  proper 
fraction ;  and  qn  is  also  a  fraction,  which  diminishes  as  n 
increases.     Therefore,  the  greater  the  number  of  terms  we 

take,  the  more  will X  qn    diminish,  and  consequent- 
ly the  more  will  the  entire  sum  of  all  the  terms  approximate 

to   an   equality   with  the  first  part  of  S,  that  is,  to . 

Finally,  when  n  is  taken  greater  than  any  given  number,  or 
n  —  infinity,  then x  qn    will  be  less  than  any  given 

number,  or  will  become  equal  to  0  ;  and  the  expression 

will  then  represent  the  true  value  of  the  sum  of  all  the  terms  of 
the  series.     Whence  we  may  conclude,  that  the  expression 

12* 


274  ELEMENTARY     ALGEBRA. 

for  the  sum  of  the  terms  of  a  decreasing  progression,  in  which 
the  number  of  terms  is  infinite,  is 

S 


1-9 

that  is,  equal  to  the  first  term  divided  by  1  minus  the  ratio. 

This  is,  properly  speaking,  the  limit  to  which   the  partial 
sumt  approach,  as  we  take  a  greater  number  of  terms  in  the 

progression.     The  difference  between  these  sums  and 


may  be  made  as  small  as  we  please,  but  will  only  become 
nothing  when  the  number  of  terms  is  infinite. 

EXAMPLES. 

1.  Find  the  sum  of 

1111 
1:   3:9  :27:8l    t0mfim^ 

We  have  for  the  expression  of  the  sum  of  the  terms 

S  = = =  —   Ans. 

3 

The   error  committed  by  taking  this  expression  for  the 
value  of  the  sum  of  the  n  first  terms,  is  expressed  by 
a 


x?B  =  f(iT' 


\-q 

First  take  n  =  5  ;  it  becomes 

1  1 

2  .   3*  ~  162 ' 


W 


171.   When  the  progression  is  decreasing  and  the  number  of  terms  in- 
finite, what  is  the  expression  for  the  valve  of  the  sum  of  the  series  ? 


GEOMETRICAL     PROGRESSION.  275 

When  n  =  6,  we  find 

3  /  1  \s  _      1  1         J_ 

2  V  3  /   ~  T02  X  3  ~  480* 

Hence,  we  see,  that  the   error  committed,  by   taking  — 

for  the  sum  of  a  certain  number  of  terms,  is  less  in  propor- 
tion as  this  number  is  greater. 

2.  Again,  take  the  progression 

,.11111. 

1  :   2   :    4   :    8   :  16  :  32  :  &C'  '  *  ' 

W  e  have         S  = = —  =  2.     Am. 

1 —9       I- — 
2 

3.  What  is  the  sum  of  the  progression 

X'W  m  1555'  ioooo'  &Mo  infinity. 

8=*-=        *         =ll     An, 
1-3  J_  <J 

10 

172.   In  the  several   questions  of  geometrical  progression 
there  are  five  numbers  to  he  considered  : 

1st.  The  first  term, a. 

2d.    The  ratio,       ....          q. 

3d.    The  number  of  terms, n. 

4th.  The  last  term, I. 

5th.  The  sum  of  the  terms, S. 


172.  How  many  numbers  are  considered  in  geometrical  profession' 
What  are  they? 


276  ELEMENTARY    ALGEBRA. 

173.  We  shall  terminate  this  subject  by  solving  this 
problem. 

To  find  a  mean  proportional  between  any  two  numbers, 
as  m  and  n. 

Denote  the  required  mean  by  x.  We  shall  then  have 
(Art.  156), 

x2  =       m  X  w, 


and  hence,  x   =z  -y/  m  X  n. 

That  is,  Multiply  the   two  numbers  together,  and  extract  the 

square  root  of  the  product. 

1.  What  is  the  geometrical  mean  between  the  numbers 
2  and  8  1 


Mean  =  -/8  X  2  =  -/IB"  =  4.  Ans. 

2.  What  is  the  mean  between  4  and  16  1  Ans.  8. 

3.  What  is  the  mean  between  3  and  27?  Ans.  9. 

4.  What  is  the  mean  between  2  and  72?  A?is.  12. 

5.  What  is  the  mean  between  4  and  64  1  Ans.  16. 

173.  How  do  you  find  a  mean  proportional  between  two  numbers  ? 


OF     LOGARITHMS  277 


CHAPTER  VIII. 

Of  Logarithms. 

174.  The  nature  and  properties  of  the  logarithms  in  com- 
mon use,  will  be  readily  understood,  by  considering  atten- 
tively the  different  powers  of  the  number  10.     They  are, 
10°=  1 

101  =  10 

102  -  100 

103  =  1000 
10*  =  10000 
105  =  100000 
&c.  &c. 

It  is  plain  that  the  exponents  0,  1,  2,  3,  4,  5,  &c,  form  an 
arithmetical  series  of  which  the  common  difference  is  1  ; 
and  that  the  numbers  1,  10,  100,  1000,  1 0000,  100000,  &c, 
form  a  geometrical  progression  of  which  the  common  ratio  is 
10.  The  number  10,  is  called  the  base  of  the  system  of  loga- 
rithms ;  and  the  exponents  0,  1,  2,  3, 4,  5,  &c,  are  the  loga- 

174,  What  relation  exists  between  the  exponents  1,  2,  3,  <fcc.  ?  How 
are  the  corresponding  numbers  10,  100,  1000?  What  is  the  common 
difference  of  the  exponents  ?  What  is  the  common  ratio  of  the  corres- 
ponding numbers?  What  is  the  base  of  the  common  sjstem  of  loga- 
rithms ?  What  are  the  exponents  ?  Of  what  number  is  the  exponent 
1  the  logarithm  ?    The  exponent  2  ?     The  exponent  3  I 


278  ELEMENTARY     ALGEBRA. 

rithras  of  the  numbers  which  are  produced  by  raising  10  to 
*he  powers  denoted  by  those  exponents. 

175.  If  we  denote  the  logarithm  of  any  number  by  to, 
then  the  number  itsejf  will  be  the  mih  power  of  10  :  that  is, 
if  we  represent  the  corresponding  number  by  M, 

10m  =  M. 
Thus,  if  we  make  m  =  0,  M  will  be  equal  to  1  ;  if  ra  =  1, 
M  will  be  equal  to  10,  &c.     Hence, 

The  logarithm  of  a  number  is  the  exponent  of  the  power  to 
which  it  is  necessary  to  raise  the  base  of  the  system  in  order 
to  'produce  the  number. 

176.  Letting,  as  before,  10  denote  the  base  of  the  system 
if  logarithms,  m  any  exponent,  and  M  the  corresponding 

number  :  we  shall  then  have, 

10m  =  M 
in  which  m  is  the  logarithm  of  M. 

If  we  take  a  second  exponent  re,  and  let  N  denote  the 
corresponding  number,  we  shall  have, 

10n  =  ^V 
in  which  n  is  the  logarithm  of  N. 

If  now,  we  multiply  the  first  of  these  equations  by  the 
second,  member  by  member,  we  have 

10m  X  10D  =  10m  +  n  =  MX  xV; 
bat  since   10   is  the  base  of  the  system,  m  -f-  n  is  the  loga- 
rithm M  X  iV ;  hence, 


175.-  Tf  we  denote  the  base  of  a  system  by  10,  and  the  exponent  by 
m.  wlmt  will  represent  the  corresponding  number?  Whai  is  the  loga- 
rithm i if  a  number  I 

17(j.  To  what  is  the  sum  of  the  logarithms  of  any  two  numbers  ecpual ! 
To  what  then,  will  the  addition  of  logarithms  correspond  ? 


OF     LOGARITHMS.  279 

The  sum  of  the  logarithms  of  any  two  numbers  is  equal  to 
the  logarithm  of  their  product. 

Therefore,  the  additiofi  of  logarithms  corresponds  to  the 
multiplication  of  their  numbers. 

177.  If  we  divide  the  equations  by  each  other,  oember 
by  member,  we  have, 

10m_  _M_ 

but  since  10  is  the  base  of  the  system,  m  —  n  is  the  lcga- 

M 

rithmof  —  :  hence, 

If  one  number  be  divided  by  another,  the  logarithm  of  the 
quotient  will  be  equal  to  the  logarithm  of  the  dividend  dimi- 
nished by  that  of  the  divisor. 

Therefore,  the  subtraction  of  logarithms  corresponds  to  the 
division  of  their  numbers. 

178.  Let  us  examine  further  the  equations 

10°  =  1 
10'  =  10 

102  =  100 

103  =  1000 
&c.      &c. 

It  is  plain  that  the  logarithm  of  1  is  0.  and  that  the  loga- 
lithms  of  all  the  numbers  between  1  and  10,  are  greater 
than  0  and  less  than  1.  They  are  generally  expressed  by 
decimal  fractions  :  thus, 

log  2  =  0.301030. 

177.  If  one  number  be  divided  by  another,  what  will  the  logarithm 
of  the  quotient  be  equal  to?  To  what  then  wall  the  subtraction  of 
logarithms  correspond  ? 

178.  What  is  the  logarithm  of  1  ?  Between  what  limits  ate  the  Inga 
rithnis  of  all  numbers  between  1  and  111?  How  are  tLey  generally 
expressed  I 


280  ELEMENTARY     ALGEBRA. 

The  ..logarithms  of  all  the  numbers  greater  than  10  and 
less  than  100,  are  greater  than  1  and  less  than  2,  and  are 
generally  expressed  by  1  and  a  decimal  fraction  :  thus, 

log  50  =  1.G98970. 

The  part  of  the  logarithm  which  stands  on  the  left  of  the 
decimal  point,  is  called  the  characteristic  of  the  logarithm. 
The  characteristic  is  always  one  less  than  the  number  of 
places  of  figures  in  the  number  whose  logarithm  is  taken. 

Thus,  in  the  first  case,  for  numbers  between  1  and  10, 
there  is  but  one  place  of  figures,  and  the  characteristic  is  0. 
For  numbers  between  10  and  100,  there  are  two  places  of 
figures,  and  the  characteristic  is  1  ;  and  similarly  for  other 
numbers. 

Table  of  Logarithms. 

179.  A  table  of  logarithms  is  a  table  in  which  are  writ- 
ten the  logarithms  of  all  numbers  between  1  and  some 
other  given  number.  A  table  showing  the  logarithms  of 
the  numbers  between  1  and  100  is  annexed.  The  numbers 
are  written  in  the  column  designated  by  the  letter  N,  and 
the  logarithms  in  the  columns  designated  by  Log. 

How  is  it  'with  the  logarithms  of  numbers  between  10  and  100? 
What  is  that  part  of  the  logarithm  called  which  stands  at  the  left  oi 
the  characteristic  ?     "What  is  the  value  of  the  characteristic? 

179.  What  is  a  table  of  logarithms  ?  Explain  the  manner  of  finding 
the  logarithms  of  numbers  between  1  and  1  JO  ? 


OF      LOGARITHMS. 

TABLE. 


281 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0.000000 

26 

1.414973 

51 

1.707570 

76 

1.880814 

2 

0.301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0.477121 

28 

1.447158 

53 

1.724276 

78 

1.892095 

4 

0.602060 

29 

1.462398 

54 

1.732394 

79 

1.897627 

5 

~6 

0.698970 

30 
34 

4.477121 

L491362 

55 

56 

1.740363 

L748188 

80 
81 

1.9030901 
1.908485 

0.778151 

7 

0.845098 

32 

1.505150 

57 

1.755875 

82 

1.913814 

8 

0.903090 

00 
00 

1.518514 

58 

1.763428 

83 

1.919078 

9 

0.954243 

34 

1.531479 

59 

1.770852 

84 

1.924279 

10 

1.000000 

35 

1.544008 

60 

1.778151 

85 

1.929419 

11 

L0T1393 

36 

1.556303 

01 

L785330 

~86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519 

13 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

39 

1.591065 

64 

1.806180 

89 

1.949390 

15 

1.176091 

40 

1.602060 

65 

1.812913 

90 

1.954243 

16 

1.204120, 

44 

1.612784 

m 

1.819544 

91 

L959041 

17 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.2552731 

43 

1.633468 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.643453 

69 

1.838849 

94 

1.973128 

20 

1.301030 

45 

1.653213 

70 

1.845098 

95 

1.977724 

21 

1.322219 

46 

1.662758 

71 

L85L258 

96 

1.982271 

22 

1.342423 ! 

47 

1.672098 

72 

1.857333 

97 

1.986772 

23 

1.361728 

48 

1.681241 

73 

1.863323 

98 

1.991226 

24 

1.38021  ll 

49 

1.690196 

74 

1.869232 

99 

1.995635 

25 

1 .397940 ! 

50 

1 .698970 

75  1.875061 

100 

2.000000 

EXAMPLES. 

1.  Let  it  be  required  to  multiply  8  by  9,  by  means  of  loga- 
rithms. We  have  seen,  Art.  176,  that  the  sum  of  the  loga- 
rithms is  equal  to  the  logarithm  of  the  product.  Therefore, 
find  the  logarithm  of  8  from  the  table,  which  is  0.903090. 
and  then  the  logarithm  of  9,  which  is  0.954243  ;  and  their 
sum,  which  is  1.857333,  will  be  the  logarithm  of  the  product. 

13 


2S2  ELEMENTARY     ALGEBRA. 

In  searching  along  in  the  table,  we  find  that  72  stands  oppo 
site  this  logarithm  :  hence,  72  is  the  product  of  8  by  9C 

2.   What  is  the  product  of  7  by  12] 

Logarithm  of  7  is,     .         .         .         .     0.845098 
Logarithm  of  12  is,    ....     1.079181 


Logarithm  of  their  product,         .  .      1.924279 

and  the  number  corresponding  is  84.  

3.  What  is  the  product  of  9  by  11 1 

Logarithm  of  9  is,      .         .         .  .     0.954243 

Logarithm  of  11  is,   .         .         .  .     -1.041393 


Logarithm  of  their  product,        .         .     1.995636 
and  the  corresponding  number  is  99. 

4.  Let  it  be  required  to  divide  84  by  3.  We  have  seen 
in  Article  177,  that  the  subtraction  of  Logarithms  corres- 
ponds to  the  division  of  their  numbers.  Hence,  if  we  find 
the  logarithm  of  84,  and  then  subtract  from  it  the  logarithm 
of  3,  the  remainder  will  be  the  logarithm  of  the  quotient. 
The  logarithm  of  84  is,  .  .  .  1.924279 
The  logarithm  of  3  is,        .         .         .     0.477121 


Their  difference  is,               .         .  .     1.447158 

and  the  number  corresponding  is  28.  ■ 

5.  What  is  the  product  of  6  by  7  1 

Logarithm  of  6  is,      .         .         .  .     0.778151 

Logarithm  of  7  is,     .         .         .  .     0.845098 


Their  sum  is, 1.623249 

and  the  corresponding  number  of  the  table,  42. 


SUPPLEMENT. 

EXAMPLES     IN     ADDITION     AND     SUBTRACTION. 

1.  What  is  the  sum  of 

axa  -f-  bx°   and   cxa  +  dxD. 

2.  What  is  the  sum  of 

ax*   and    bxn  —  cxa  —  dxn. 

3.  What  is  the  sum  of 

10a4  +  3a4   and   6a*  —  a*  —  5a*. 

4.  What  is  the  sum  of 

5a3  —  7a3   and    11a3  +  a4. 

5.  What  is  the  sum  of 

an6m  —  9am  +  5an6m   and   6am  +  10an6ra. 

6.  What  is  the  sum  of 

ba?b2+7ab2c— Samb5  —  12ab2c    and    6a3b2  — SaW  +  b'SaTb1 
-362. 

7.  What  is  the  sum  of 

5a46  +  3a2b2c  —  lab  —  6a*b    and    2«262c  +  17a5  -f  9a46  - 
8a262c  —  10a6. 

8.  What  is  the  sum  of 

5am6P  -f  Sa3^-1  —  3a3  —  3cara6p    and    ^Vi™-1  —  a  +  10a3 
+  am6p  +  a  +  3a2&2  --  2g2a3bm~\ 

9.  What  is  the  sum  of 

9a3b2c*  —  lb  +  186  —  banbm  -f  c*  —  3c/5  and  3an6m  —  haWi* 
+  3c1  —  5d5. 

10.  From     —  9amx2  —  13  +  2ab3x  —  4bmcx2 
take      36rac.7:2  —  9amz2  —  C  +  2a&3*. 


284      ELEMENTARY     ALGEBR  A S  UPPLEMENT, 

11.  From  5a*  -  laW  —  2>cd2  +  Id 
take  —  'iha?b2  +  3a4  —  3a2  —  led2. 

12.  From  QaH*  +  6abm  —  d5  +  18a*6° 
take  7a*bD  +  cZ5  —  Sa6m  +  9amZ>2. 

13.  From  12tV6  —  16a8£6  —  5amZ/n  +  Gacb 
take  6am6n  —  Gacb  +  lGa8Z>6  +  1265<Z6. 

il  From  8a3b3c5  —  Uamb  +  Qax*  +  8a^m 

take  8«fZm  —  $a3b3c5  —  12amb  +  Gax*. 

15.  From  12amJn  —  9az5  —  Aab  +  6a2Z>2  —a 

take  3a  —  6a2Z>2  +  12amZ>n  —  9ax5  +  5a&, 


1.  What 


2.  What 


3.  What 


4.  What 


EXAMPLES    IN    MULTIPLICATION'. 

s  the  simplest  form  of  the  product  of 

am  X  an. 
s  the  simplest  form  of  the  product  of 

2a3  x  7a9  x  —  3a8. 
s  the  simplest  form  of  the  product  cf 
a5b  X  d!d  X  10a  X  &a2  X  —  1. 
s  the  simplest  form  of  the  product  of 


_  ap-q  x  —  3a?~2  X  /  X  5a3<H-7cr. 

5.  What  is  the  simplest  form  of  the  product  of 

5a3£*  x  10a255c  X  —3a7. 

6.  What  is  the  simplest  form  of  the  product  of 

—  7a568c2  x  Sa5b2d  x  7b8c  X  —  1. 

7.  What  is  the  simplest  form  of  the  product  of 

amb*cq  X  effrc"1  X  amb  X  —  a. 

8.  What  is  the  simplest  form  of  the  product  of 

(a2  —  3a5  —  5b2)  X  4a?b. 

9.  What  is  the  simplest  form  of  the  product  of 

(2a3Z,5  -  5aV  4-  9a3o2c3)  X  3a2ic2. 


EXAMPLES     IN     MULTIPLICATION.  2S5 

10.  What  is  the  simplest  form  of  the  product  of 

{IhH  +  2P  —  3aA3/2  +  7)  X  -  8JcH5. 

11.  What  is  the  simplest  form  of  the  product  of 

(a364  _  cj*dy+  3c-)  x  _  2bc2d. 

12.  What  is  the  simplest  form  of  the  product  of 

13.  What  is  the  simplest  form  of  the  product  of 

(3F  -  5kl  +  2P)  X  (P  -  Vcl). 

14.  What  is  the  simplest  form  of  the  product  of 

(6/2  -  17/7  4-  3/2)  x  (/5  +  4/H). 

15.  What  is  the  simplest  form  of  the  product  of 

(4a2  —  lQax  +  3z2)  X  (5a3  —  2a2x). 

16.  What  is  the  simplest  form  of  the  product  of 

(a2  +  a4  4-  a6)  X  (a2  —  1). 

17.  What  is  the  simplest  form  of  the  product  of 
(a4  -  2a36  4-  4a262  —  8aP  4-  1664)  X  (a  4-  26). 

18.  What  is  the  simplest  form  of  the  product  of 

(2a%2  -  36«y8)  X  (2a4x2  +  3&4r/2). 

19.  What  is  the  simplest  form  of  the  product  of 
(7a3  —  5a26  4-  Qab2  -  2b3)  X  (3a4  —  4a3b  4-  16a262). 

20.  What  is  the  simplest  form  of  the  product  of 

(a6  —  3a462  4-  5a264)  X  (7a4  -  4a262  4-  b*). 

EXAMPLES    IN    DIVISION,, 

1.  Divide  ara  by  a". 

2.  Divide  am  by  a2". 

3.  Divide  8a16  by  2a*. 

4.  Divide  calB  by  aa4. 


286   ELEMENTARY  ALGEBR  A S  UPPLEMENT. 


5.  Div 

6.  Div 

7.  Div 

8.  Div 

9.  Div- 
10.  Div 

11.  Div 

12.  Div 

13.  Div 


ide  G(a  +  bf  by  3(a  +  bf. 

ide  (a  +  x)*  x  (a  +  yf   by    (a  +  x)  x  (a  +  y)\ 

ide  Ga3&2  —  loa'^'-f-  27 a*bx,   by  oul 

ide  be3  —  c3x  by  &  —  x. 

ide  a3  +  a2b  —  ab2  —  b3  by  a  —  6. 

ide  3a5  +  16a46  -  33a362  +  14a263  by  a2  +  lab. 

ide  a7  —  6«663  +  14a5i6  —  12a*i9  by  a3  -  2a253. 

ide  a*  —  2a262  +  b*  by  a2  —  62. 

ide   —  a%*  +  lSa1^5  -  48au65  —  20a1767 
by   10a962  — a66. 

14.  Divide  a8  —  I628  by  a2  —  2z2. 

15.  Divide  2a*  -  13a36  +  31a262  -  38ai3  +  2^)* 
by  2a2  —  Sab  +  452. 

1(5.  Divide  4c4  -  962c2  +  6b3c  —  i4  by  2c2  —  3bc  4-  62. 

17.  Divide  —  1  +  a3n3  by   —  1  4-  an' 

18.  Divide  a6  4"  2a323  4-  s6  by  a2  —  az  -\-  z~. 

19.  Divide  i  —  6z2  4-  27z"  by  ^  4-  2z  +  S*2. 

20.  Divide  a6  —  10a3z3  4-  G4.z6  by  a2  —  4az  4-  4z2. 

21.  Divide  aid3  —  3a2cd3  +  3ac2d3  —  c3d3  +  a2c2d2  —  ac^oP 
by  a?d2  —  2acd2  +  c2cP  4  ac2d. 


EXAMPLES    IN    REDUCTION    OF    FRACTIONS. 

1.  Reduce  to  its  simplest  terms  the  fraction 

l&acf—  Qbdcf—  2ad 
Sadf 

2.  Reduce  to  its  simplest  terms  the  fraction 

8a2  —  Gab  4-  4c  4-  1 
—  2a  ' 


EXAMPLES    IN    REDUCTION    OF    FRACTIONS.    287 

3.  Reduce  to  its  simplest  terms  the  fraction 

Macfg  -  4apff  +  Wh 
4.a2b2/g. 

4.  Reduce  to  its  simplest  terms  the  fraction 

ab  —  ac 


b-c 


5.  Reduce   a  —  b  -\ to  the  form  of  a  fraction. 

x  —  a 

b  —  ?/ 

6.  Reduce   x to  the  form  of  a  fraction. 


7.  Reduce   a  +  b  +  — ; —  to  ►he  form  of  a  fraction. 
a 


c  —  d 
8.  Reduce   x  —  ab >  the  form  of  a  fraction. 


9.  Reduce   a to  the  form  of  a  fraction. 

x  —  y 

b  "4*  cfx 
10.  Reduce   Gaf2x-{-9af—  —r —  to  the  form  of  a  fraction, 


1 1 .  Reduce   5acx  —t to  the  form  of  a  fraction. 

fac 

12.  Reduce  to  au  er  are,  or  mixed  quantity,  the  fraction 

0%2  —  10a2f+7a4bx 
202- 

13.  Reduce  */        entire,  or  mixed  quantity,  the  fraction 

^a2x5  —  %ax3  -f-  Sabx 


288   ELEMENTARY  ALGEBRA SUPPLEMENT. 

14.  Reduce  to  an  entire,  or  mixed  quantity,  the  fraction 

a3  —  2a2x  +  ab  +  ax2  —  bx 
a  —  x 

15.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

a  —  x      b  —  c    Aax  —  c 


a  +  x  '        f    '      a  — 


x 


16.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

a  a  —  x  c 

and 


Sax  —  &       Sb  a  —  x' 

17.  Reduce  the  following  fractions  to  a  common  denomi 
nator :  viz. 

Ao if —  x        a  —  x        .     5ac 

-~~ — , and   . 

la  —  c  c  y 

18.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

Aax  +  b         .    8ac  —  f 

5 and    — - — J-. 

Sac  —  f  Aax 

19.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

a  +  x      a  —  b               c  —  d 
and    . 


a  —  x"1    a  +  x 

20.  Reduce  the  following  fractions  to  a  common  denomi 
nator :  viz. 

a  4-  b  —  c     c  -\-f        ,     x  —  a 

,    —    and    . 

x  —  a  c  x  -{-  a 


EXAMPLES    IN    ADDITION    AND    SUBTRACTION.    289 


ADDITION,    SUBTRACTION,    MULTIPLICATION,    AND    DIVISION    OP 

FRACTIONS. 

1.  What  is  the  sum  of , and  cl 

x  —  a     a  —  x 


2.  What  is  the  sum  of  — - — , and  yl 

b     ' a— c  * 

«    nr,       .     ,  .  3a     z  —  ay       Aax    „ 

3.  What  is  the  sum  of  — ,    — —-, 1 

b  '         d      '   8c  —  / 

,    ttt,       .1  ~  ax  —  f    ac  —  a     m      n 

4.  What  is  the  sum  of -,    —,    5ayl 

c  +  a'       —  x  '         v 

QctX  '■  »■  A? 

5.  What  is  the  sum  of   8a  -\ ,  2a  —  -  ] 

x  —  a  a-\-b 

n    ^  ,    3a       .      6a  —  x 

6.  x  rom  8a  +  -7-  take . 

o  a  —  # 

_,    _  8x  —  ax      .  5az 

7.  From  p—   take  3 . 

b  c 

_    _  aa;  +  3ay  J   .      Saa;  —  0/ 

8.  From take ^. 

acx  —  ay  ay 

^    -n  ffit  +  tf",,      „        ,   6ax  —  y 

9.  From  ay take  Say  -f- 


9 


10.  From  -^  take  7«6*  -  ^^. 

8a:  8a  —  a; 

3&  a  —  x 

11.  Multiply  7a  +  —  by    — ; — . 

r  J  0      J    x  +  a 

.  ^    -.«-,.  ,         5a      ,      n  5ax  —  z2 

12.  Multiply    by  Say ■ 

13 


290       ELEMENTARY     ALGEBRA — SUPPLEMENT. 


,«,.-,.,      9a  —  x  ,       _.  6az  —  x2 

13    Multlply~~by2a+— -. 


14.  Multiply  Sax  +  ^L_£   by  5  +  4-. 

,,-»*■!•!      />      ,5  —  Sa,       6a  —  b 

15.  Multiply   6a  H r    by 


—  6 


a2  —  62 


16.  Divide  3ac  —  2adc  —  f  + -j  by  2a. 

17.  Divide  ~  +  4"  -  3«c  +  7   by  ^-. 

a         2c  J     a 


18.    Divide  3a3  - 
by   3a  -  56  +  --. 


lab         2]ac        562        836c  _  3e» 
"2  4  2~    *    "~8  '2 


to    i  •  -a     on       55/ft      2!)/i,    21/t»       ISA*   ,    ** 
19.  1  ivide   y1  --_  +  _  +  _  — j-  +  3- 


u    2/       SA  , 


„    _     , ,              5x2           llrw          10.rg  15?/2    ,    ,.„ 

20.  Divide —  +  -^- —  +  -~  +  ^yt 

2x 

oy  —-g  +  Sy- 


EXAMPLES    IN    EQUATIONS    OF    THE    FIRST    DEGREE. 


1.  Given   * 


*-lv  +  ^  =  H 


x  —  y 


>  to  find  x  and  y. 


+  7x  =  41 


EQUATIONS     OF     THE     FIRST     DEGREE.        291 

f  x—y  , ®+y 


2.  Given   <j        4       +     5  *  ~    ^ 


to  find  £  and  y. 


3.  Given  ax  -\ 1-  |  (x  — /)  =  y  to  find  a;. 

_.         a  —  x       b  —  x       a  —  x        .       .•_    , 

4.  Given  — : ( =  a,  to  find  *. 


4  5 

3y  —  a;      2a;  —  y 


5.  Given 


G 


+ 


4 

3 -2a; 
4 


ss    5 


to  find  x  and  y. 


6.  Given  < 


6*-y4        —  =  43i 
3ar  — 8      y  —  6 


J 


+ 


+  y  =  i8^ 


ra 


8a;  _  3  -  ^-^  =  79 


7.  Given  < 


4a;  — 4      y  —  5   ,   A       ,__ 
3 ?L_-  +  6=12| 


**-*y+nr 


to  find 
x  and  y. 


to  find 
x  and  y. 


a  — a;       a  — 2a;  . 

8.  Given   — = h  «  =  6,  to  find  a;. 


_     ~.         3a  —  6a;       2a  —  3a;   ,  .        .        _    , 

9.  Given    — ^ h  a;  —  a  =zf,  to  find  x. 


10.  Given    ■{  ,        "  ,  [    to  find  x  and  y. 

(a  —  y  -\-  x  ■=.  a  ) 

11.  Given  ^--  +  ^i  +  4  (as  -  3)  =  G8,  to  find  x. 

O  6 


292   ELEMENTARY  ALGEBRA SUPPLEMENT. 


12.  Given    < 


o 

Z — X 


►    to  find  x,  y  and  z. 


x  +  2y  +  3z  =  14 
13.  Given     -{  z  —  y  +  2  =  2         }■   to  find  a;,  y  and  z. 
3x4-  6y +  z  =  18 


14.  Given    < 


rl*-'*y  +  **  =  2J  - 

3 
10 


^  +  1+5  =  4* 


a;  —  3       y  —  z 

2^4 


>    to  find  a?,  y  and  z. 


..     «.  j  13s4-7y— 341  =74y  4-431*1    -    .    ,  , 

Id.  Given     \  n     .  ,  *   ,  2*         2    >-  to  find  x  and  y. 

16.  Given     H-+5)(y-f  7)=(,:+l)(y-9)+112  )     to  find 

(  2x-fl0  =  3y-}-l  jxandy. 


17.  Given 


{." 


by 

+  y  =e 


to  find  x  and  y. 


18.  Given     \  ~         ^       ,  {•  to  find  x  and  y. 

(  -JL.  =  _A_  ) 

19.  Given     ■<  j  _|_  »        3a  _f_  .T  >   to  find  x  and  y. 

(  a.r  +  2  by  =  c/      J 

hex  =  cy  —  26 

20.  Given     J         ,  o(f3  -  63)      2J3   ,     ,    \  to  find*  and  y. 

&2y  4 — - — = 1-  c3x  ' 


11.  Given    < 


3x  4-  5y  = 


(85  -  2f)bf 


y-x=- 


J2  _/2 

—  25/'7 


62— /» 


to  find  x  and  y. 


EQUATIONS     OF     THE     FIRST     DEGREE.        293 

j  x  +  y  +  z  =  29^  \ 
22.  Given    )  x  +  y  —  z  =  18^  I  to  find  a;,  y  and  z. 
\x-y  +  z  =  \Zl) 

(3x  +  5y  =  161  ) 
23    Given    -J  7x  +  2z  —  209  V  to  find  x,  y  and  z. 
( 2y  +    z  =  89    ) 


24.  Given   < 


r  1 

1 

+ 

r= 

a 

X 

y 

1 

i 

+ 

z=. 

b 

a; 

z 

1 

1 

— 

+ 

— 

— 

c 

1  y 

2 

to  find  x,  y  and  z. 


{ax  -\-  by  =z  c  \ 
dx  +  ey  =f>  to  find  x%  y  and  2. 
#y  +  hz  =  ^  ) 


EXAMPLES    IN    EQUATIONS    OF   THE    SECOND    DEGREE. 

1.  Given  x2  —  5\x  =  18  to  find  x. 

2.  Given  3x2  —  2x  =  65  to  find  x. 

3.  Given  622x  =  15x2  +  6384  to  find  a?j 

4.  Given  11  J*  —  3£z2  =  —  41},  to  find  x. 

5.  Given  9£z2  —  90^  +  195  =  0,  to  find  x. 

6.  Given  20748  —  1616*  +  21*2  =  0,  to  find  x. 

7.  Given  9l3x2  —  90£x  +  195  =  0,  to  find  x. 

8.  Given     ^  +   ™™1  +  4728  =  0,  to  find  X. 

5  05 


294   ELEMENTARY  ALGEBRA SUPPLEMENT. 

9.  Given  x2  —  8*  =  14    to  find  *. 

10.  Given  ox2  +  x  =  7   to  find  x. 

11.  Given  118s  —  2i*2  —  20   to  find  x. 

12.  Given  Gx  —  30  =  3*2   to  find  x. 

13.  Given  8x2  —  7*  +  34  =  0   to  find  x. 

14.  Given  4.x2  —  9*  =  5*2  —  255^  —  8a;   to  find  x. 

,.    ^.  ™      "    ox2      21*  — 27782       ,„„rt, 

15.  Given    80*  +  — -  -\ -- =  1S59J  —  3** 

4  12  3 

1G.  Given     —  =  — — —  -+-  1    to  find  x. 

6*       117  —  2* 

17.  Given     ^±i^  =  J**L  _  1   to  find  * 

10* -81        5* -8        5 

1Q  18  +  *         20* +  9  65 

18.  Given     — r- r  =  — —j^ =    to  find  «. 

6(3  —  *)       19  —  7*      4(3  —  *) 

19.  Given     *2  —  7*  +  3±  =  0   to  find  *. 

20.  Given     4*2  —  9*  =  5*2  -  255 \  -  8*   to  find  x. 

*  7 

21.  Given     — t-xtt  =  « =-  to  find  *. 

*  -f  60        3*  —  5 

40  27 

22.  Given -\ =13   to  find  *. 

*  —  5         * 

23.  Given    — ^-  -  6  =  ~  to  find  *. 

*  +  2  3* 

...    ^.              48              165  .   _ 

H4.  Given — -  = — -  —  5   to  find  *. 

*  +  3       *  -{-  10 


PROMISCUOUS  PROBLEMS, 

GIVING   EISE  TO 

EQUATIONS   OF   THE   FIRST   DEGREE. 

1.  A  person  expended  30  cents  fur  apples  and  pears, 
giving  one  cent  for  four  apples,  and  one  cent  for  live  pears: 
he  then  sold,  at  the  prices  he  gave,  half  his  apples  and  one- 
third  his  pears,  for  13  cents.  How  many  did  he  buy  of 
each  ? 

2.  A  tailor  cut  10  yards  from  each  of  three  equal  pieces 
of  cloth,  and  17  yards  from  another  of  the  same  length, 
and  found  that  the  four  remnants  were  together  equal  to  142 
yards.     How  many  yards  in  each  piece? 

3.  A  fortress  is  garrisoned  by  2000  men,  consisting  of 
infantry,  artillery,  and  cavalry.  Now,  there  are  nine  times 
as  many  infantry,  and  three  times  as  many  artillery  soldiers, 
as  there  are  cavalry.     How  many  are  there  of  each  corps'? 

4.  All  the  journeyings  of  an  individual  amounted  to  2970 
miles.  Of  these  he  travelled  3^  times  as  many  by  water 
as  on  horseback,  and  2-1  times  as  many  on  foot  as  by  water. 
Ilow  many  miles  did  he  travel  in  each  way  % 

5.  A  sum  of  money  was  divided  between  two  persons, 
A  and  B.  A's  share  was  to  B's  in  the  proportion  of  5  to  3, 
and  exceeded  five- ninths  of  the  entire  sum  by  50.  What 
was  the  share  of  each1? 

6.  There  are  52  pieces  of  money  in  each  of  two  bags,  out 
of  which  A  and  B  help  themselves.    A  takes  twice  as  much 


296   ELEMENTARY  ALGEBRA SUPPLEMENT. 

is  B  leaves,  and  B  takes  seven  times  as  much  as  A  leaves. 
How  much  does  each  take  1 

7.  Two  persons,  A  and  B,  agree  to  purchase  a  house  to- 
gether, worth  $1200.  Says  A  to  B,  give  me  two-thirds  of 
your  money  and  I  can  purchase  it  alone ;  but,  says  B  to  A, 
if  you  give  me  three-fourths  of  your  money  I  shall  be  able 
V  purchase  it  alone.     How  much  had  each  % 

8.  A  father  directs  that  $1170  shall  be  divided  among 
nis  three  sons,  in  proportion  to  their  ages.  The  oldest  is 
twice  as  old  as  the  youngest,  and  the  second  is  one-third  older 
than  the  youngest.     How  much  was  each  to  receive  ? 

9.  Three  regiments  are  to  furnish  594  men,  and  each  to 
furnish  in  proportion  to  its  strength.  Now,  the  strength  of 
the  first  is  to  the  second  as  3  to  5  ;  and  that  of  the  second 
to  the  third  as  8  to  7.     How  many  must  each  furnish  ? 

10.  A  grocer  finds  that  if  he  mixes  sherry  and  brandy  in 
the  proportion  of  2  to  1,  the  mixture  will  be  worth  78s.  per 
dozen  ;  but  if  he  mixes  them  in  the  proportion  of  7  to  2,  he 
can  get  79s.  a  dozen.  What  is  the  price  of  each  liquor  per 
dozen  1 

11.  A  person  bought  7  books,  the  prices  of  which  were  in 
arithmetical  progression,  (in  shillings).  The  price  of  the  one 
next  above  the  cheapest,  was  8  shillings,  and  the  price  of  the 
dearest,  23  shillings.     What  was  the  price  of  each  book  ? 

12.  A  number  consists  of  three  digits,  which  are  in  arith- 
metical proportion.  If  the  number  be  divided  by  the  sum 
of  tne  digits,  the  quotient  will  be  26 ;  but  if  198  be  added 
to  it,  the  order  of  the  digits  will  be  inverted. 

13.  A  person  has  three  horses,  and  a  saddle  which  is  worth 
$220.     If  the  saddle  be  put  on  the  back  of  the  first  horse,  it 


EQUATIONS     OF     THE     FIRST     DEGREE.  29? 

will  make  his  value  equal  to  that  of  the  second  and  third ; 
if  it  be  put  on  the  back  of  the  second,  it  will  make  his  value 
double  that  of  the  first  and  third ;  if  it  be  put  on  the  back 
of  the  third,  it  will  make  his  value  triple  that  of  the  first 
and  second.     What  is  the  value  of  each  horse  1 

14.  The  crew  of  a  ship  consisted  of  her  complement  of 
sailors,  and  a  number  of  soldiers.  There  were  22  sailors  to 
every  three  guns,  and  10  over  ;  also,  the  whole  number  of 
hands  was  five  times  the  number  of  soldiers  and  guns  to- 
gether. But  after  an  engagement,  in  which  the  slain  were 
one-fourth  of  the  survivors,  there  wanted  5  men  to  make  13 
men  to  every  two  guns.  Required,  the  number  of  guns, 
soldiers,  and  sailors. 

15.  Three  persons  have  $96,  which  they  wish  to  divide 
equally  between  them.  In  order  to  do  this,  A,  who  has  the 
most,  gives  to  B  and  C  as  much  as  they  have  already :  then 
B  divides  with  A  and  C  in  the  same  manner,  that  is,  by 
giving  to  each  as  much  as  he  had  after  A  had  divided  with 
them :  C  then  makes  a  division  with  A  and  B,  when  it  is 
found  that  they  all  have  equal  sums.  How  much  had  each 
at  first  1 

16.  To  divide  the  number  a  into  three  such  parts,  that 
the  first  shall  be  to  the  second  as  m  to  n,  and  the  second  to 
the  third  as  p  to  q, 

17.  Five  heirs,  A,  B,  C,  D  and  E,  are  to  divide  an  inher- 
itance of  $5000.  B  is  to  receive  twice  as  much  as  A,  and 
$200  more ;  C  three  times  as  much  as  A,  less  $400 ;  D  the 
half  of  what  B  and  C  receive  together,  and  150  more  ;  and 
E  the  fourth  part  of  what  the  four  others  get,  plus  $475. 
How  much  did  each  receive? 

18.  A  person  has  four  casks,  the  second  of  which  being 

13* 


208       ELEMENTARY      ALGEBRA SUPPLEMENT. 

filled  from  the  first,  leaves  the  first  four-sevenths  full.  The 
third  being  filled  from  the  second,  leaves  it  one-fourth  full, 
and  when  the  third  is  emptied  into  the  fourth,  it  is  found  to 
fill  only  nine-sixteenths  of  it.  But  the  first  will  fill  the  third 
and  fourth,  and  leave  15  quarts  remaining.  How  many 
quarts  does  each  hold  ? 

19.  A  courier  having  started  from  a  place,  is  pursued  by 
a  second  after  the  lapse  of  10  days.  The  first  travels  4 
miles  a  day,  the  other  9.  How  many  days  before  the 
second  will  overtake  the  first  ? 

20.  If  the  first  courier  had  left  n  days  before  the  other, 
and  made  a  miles  a  day,  and  the  second  courier  had  travelled 
b  miles,  how  many  days  before  the  second  would  have  over- 
taken the  first  1 

21.  A  courier  goes  31  J  miles  every  five  hours,  and  is  fol- 
lowed by  another  after  he  had  been  gone  eight  hours.  The 
second  travels  22^  miles  every  three  hours.  How  many 
hours  before  he  will  overtake  the  first  % 

22.  Two  places  are  eighty  miles  apart,  and  a  person  leaves 
one  of  them  and  travels  towards  the  other,  at  the  rate  of  3| 
miles  per  hour.  Eight  hours  after,  a  person  departs  from  the 
second  place,  and  travels  at  the  rate  of  5^  miles  per  hour. 
How  long  before  they  will  meet  each  other  ] 

23.  Three  masons,  A,  B  and  C,  are  to  build  a  wall.  A 
and  B  together  can  do  it  in  12  days  ;  B  and  C  in  20  days ; 
and  A  and  C  in  15  days.  In  what  time  can  each  do  it  alone, 
and  in  what  time  can  they  all  do  it  if  they  work  together? 

24.  A  laborer  can  do  a  certain  work  expressed  by  a,  in  a 
time  expressed  by  b  ;  a  second  laborer,  the  work  c  in  a  time 
d;  a  third,  the  woik  e  in  a  time/     It  is  required  to  find  the 


EQUATIONS     OF     THE     FIRST     DEGREE.  299 

lime  it  would  take  the  three  laborers,  working  together,  to 
perform  the  work  <j. 

25.  Required  to  find  three  numbers  with  the  following 
conditions.  If  0  be  added  to  the  1st  and  2d,  the  sums  are 
to  one  another  as  2  to  3.  If  5  be  added  to  the  1st  and  3d, 
the  sums  are  as  7  to  1 1  ;  but,  if  86  be  subtracted  from  the 
2d  and  3d,  the  remainders  will  be  as  0  to  7. 

20.  The  sum  of  $300  was  put  out  at  interest,  in  two 
separate  sums,  the  smaller  sum  at  two  per  cent,  more  than 
the  other.  The  interest  of  the  larger  sum  was  afterwards 
Increased,  and  that  of  the  smaller  diminished  by  one  per 
jent.  By  this,  the  interest  of  the  whole  was  augmented  one- 
iburth.  But  if  the  interest  of  the  greater  sum  had  been  so 
increased,  without  any  diminution  of  the  less,  the  interest  of 
the  whole  would  have  been  increased  one-third.  What  were 
the  sums,  and  what  the  rate  per  cent.  1 

27.  The  Ingredients  of  a  loaf  of  bread  weighing  \5lbs.,  are 
rice,  flour  &nJ  water.  The  weight  of  the  rice,  augmented  by 
Gibs.,  is  two-thirds  the  weight  of  the  flour  ;  and  the  weight 
of  the  water  is  one-fifth  the  weight  of  the  flour  and  rice 
together.     Rec[uired,  the  weight  of  each. 

28.  Several  detachments  of  artillery  divided  a  certain  num- 
ber of  cannon  balls.  The  first  took  72  and  £  of  the  remain- 
der ;  the  next  144  and  I  of  the  remainder;  the  third  210 
and  £  of  the  remainder ;  the  fourth  288  and  g  of  what  was 
left ;  and  so  on,  until  nothing  remained  ;  when  it  was  found 
that  the  balls  were  equally  divided.  Required,  the  number 
»>f  balls  and  the  number  of  detachments. 

29.  A  banker  has  two  kinds  of  money  ;  it  takes  a  pieces 


300     ELEMENTARY     ALGEBRA SUPPLEMENT. 

of  the  first  to  make  a  crown,  and  b  of  the  second  to  make 
the  same  sum.  lie  is  offered  a  crown  for  c  pieces.  How 
many  of  each  kind  must  he  give  1 

30.  Find  what  each  of  three  persons,  A,  B  and  C  is 
worth,  knowing,  1st,  that  what  A  is  worth,  added  to  I  times 
what  B  and  C  are  worth,  is  equal  to  p ;  2d,  that  what  B  is 
worth,  added  to  m  times  what  A  and  C  are  worth,  is  equal 
to  q  ;  3d,  that  what  C  is  worth,  added  to  n  times  what  A 
and  B  are  worth,  is  equal  to  r. 

31.  Find  the  values  of  the  estates  of  six  persons,  A,  B, 
C,  D,  E  and  F,  from  the  following  conditions.  1st.  The  sum 
of  the  values  of  the  estates  of  A  and  B  is  equal  to  a ;  that 
of  C  and  D  to  b ;  and  that  of  E  and  F  to  c.  2d.  The 
estate  of  A  is  worth  m  times  that  of  C ;  the  estate  of  D  is 
worth  n  times  that  of  E,  and  the  estate  of  F  is  worth  p 
times  that  of  B. 


PROMISCUOUS    PROBLEMS, 
INVOLVING  EQUATIONS  OF  THE    SECOND  DEGHEE. 

1.  Find  three  numbers,  such,  that  the  difference  between 
the  third  and  second  shall  exceed  the  difference  between  the 
second  and  first  by  6 :  that  the  sum  of  the  numbers  shall  be 
33,  and  the  sum  of  their  squares  467. 

2.  It  is  required  to  find  three  numbers  in  geometrical 
progression,  such  that  their  sum  shall  be  14,  and  the  sum  of 
their  squares  84. 

3.  What  two  numbers  are  those,  whose  sum  multiplied 
by  the  greater,  gives  144,  and  whose  difference  multiplied 
by  the  less,  gives  14  % 

4.  What  two  numbers  are  those,  which  are  to  each  other 
as  m  to  n,  and  the  sum  of  whose  squares  is  b  % 

5.  What  two  numbers  are  those,  which  are  to  each  other 
as  m  to  n,  and  the  difference  of  whose  squares  is  b  1 

6.  A  certain  capital  is  out  at  4  per  cent,  interest.  If  we 
multiply  the  number  of  dollars  in  the  capital  by  the  num- 
ber of  dollars  in  the  interest,  for  five  months,  we  obtain 
$  1 1 704 1  f .     W  hat  is  the  capital  1 

7.  A  person  has  three  kinds  of  goods,  which  together  cost 
$230-257.  One  pound  of  each  article  costs  as  many  times  2V 
of  a  dollar  as  there  are  pounds  of  that  article.     Now,  he  has 


302    ELEMENTAKY  A1GEBRA SUPPLEMENT. 

one-third  more  of  the  second  kind  than  of  the  first,  and  3J 
times  more  of  the  third  than  of  the  second.  How  many 
pounds  had  he  of  each  1 

8.  Required  to  find  three  numbers,  such,  that  the  product 
of  the  first  and  second  shall  be  equal  to  a ;  the  product  of 
the  first  and  third  equal  to  b ;  and  the  sum  of  the  squares 
of  the  second  and  third  equal  to  c. 

9.  It  is  required  to  find  three  numbers,  whose  sum  shall 
be  38,  the  sum  of  their  squares  634,  and  the  difference  be- 
tween the  second  and  first  greater  by  7  than  the  difference 
between  the  third  and  second. 

10.  Find  three  numbers  in  geometrical  progression,  whose 
sum  shall  be  52,  and  the  sum  of  the  extremes  to  the  mean, 
as  10  to  3. 

11.  The  sum  of  three  numbers  in  geometrical  progression 
is  13,  and  the  product  of  the  mean  by  the  sum  of  the  ex- 
tremes is  30.     What  are  the  numbers? 

12.  It  is  required  to  find  three  numbers,  such.,  that  the 
product  of  the  first  and  second,  added  to  the  sum  of  their 
squares,  shall  be  37;  and  the  product  of  the  first  and  third, 
added  to  the  sum  of  their  squares,  shall  be  49  ;  and  the  pro- 
duct of  the  second  and  third,  added  to  the  sum  of  theii 
squares,  shall  be  61. 

14.  Find  two  numbers,  such,  that  their  difference,  added 
to  the  difference  of  their  squares,  shall  be  equal  to  150.  and 
whose  sum,  added  to  the  sum  of  their  squares,  shall  be  equal 
to  330. 

15.  It  is  required  to  find  a  number  consisting  of  three 
digits,  such,  that  the  sum  of  the  squares  of  the  digits  shall  be 


EQUATIONS     OF     THE     SECOND     DKGliEE,       303 

104  ;  the  square  of  the  middle  digit  to  exceed  twice  the 
product  of  the  other  two  by  4;  and  if  594  be  subtracted 
from  the  number,  the  three  digits  become  inverted. 

16.  The  sum  of  two  numbers  and  the  sum  of  their  squares 
being  given,  to  find  the  numbers. 

17.  The  sum,  and  the  sum  of  the  cubes,  of  two  numbers 
being  given,  to  find  the  numbers. 

18.  To  find  three  numbers  in  arithmetical  progression 
such,  that  their  sum  shall  be  equal  to  18,  and  the  product 
of  the  two  extremes  added  to  25  shall  be  equal  to  the  square 
of  the  mean. 

19.  Having  given  the  sum,  and  the  sum  of  the  fourth 
powers  of  two  numbers ;  to  find  the  numbers. 

20.  To  find  three  numbers  in  arithmetical  progression, 
such,  that  the  sum  of  their  squares  shall  be  equal  to  1232, 
and  the  square  of  the  mean  greater  than  the  product  of  the 
two  extremes,  by  16. 

21.  To  find  two  numbers  whose  sum  is  80,  and  such,  that 
if  they  be  divided  alternately  by  each  other,  the  sum  of  the 
quotients  shall  be  3^. 

22.  To  find  two  numbers  whose  difference  shall  be  10, 
and  if  600  be  divided  by  each  of  them,  the  difference  of  the 
quotients  shall  also  be  10. 


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